i 


m. 


IN  MEMORIAM 
FLORIAN  CAJORI 


•*•{.. 


WENTWORTH'S 
SERIES    OF     MATHEMATICS. 


First  Steps  in  Number. 

Primary  Arithmetic. 

Grammar  School  Arithmetic. 

High  School  Arithmetic. 

Exercises  in  Arithmetic. 

Shorter  Course  in  Algebra. 

Elements  of  Algebra.  Complete  Algebra. 

College  Algebra.  Exercises  in  Algebra. 

Plane  Geometry. 

Plane  and  Solid  Geometry. 

Exercises  in  Geometry. 

PI.  and  Sol.  Geometry  and  PI.  Trigonometry. 

Plane  Trigonometry  and  Tables. 

Plane  and  Spherical  Trigonometry. 

Surveying. 

PI.  and  Sph.  Trigonometry,  Surveying,  and  Tables. 

Trigonometry,  Surveying,  and  Navigation. 

Trigonometry  Formulas. 

Logarithmic  and  Trigonometric  Tables  (Seven}, 

Log.  and  Trig.  Tables  (Complete  Edition). 

Analytic  Geometry. 


Special  Terms  and  Circular  on  Application, 


GRAMMAR   SCHOOL 


ARITHMETIC 


BY 


G.  A.  WENTWORTH,  A.M., 

i  * 
1'KnKK^soii    OK    MAYHKMATK  >    IN    I'll  I  1. 1. 1  PS    EXETER   ACADEMY. 


REVISED   EDITION. 


BOSTON,  U.S.A.: 

PUBLISHED   BY   GINN   &   COMPANY. 
1892. 


Entered,  according  to  Act  of  Congress,  in  the  year  1889,  by 

Q.  A.  WENT  WORTH, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


ALL  RIGHTS  RESERVED. 


TYPOGRAPHY  BY  J.  S.  CUSHING  &  Co.,  BOSTON,  U.S.A. 


PRESSWORK  BY  GINN  &  Co.,  BOSTON,  U.S.A. 


f  <?ca 


PREFACE. 


^F^IIIS  Arithmetic  is  designed  to  give  pupils  of  the  grammar-school 
age  an  intelligent  knowledge  of  the  subject  and  a  moderate 
power  of  independent  thought. 

Whether  Arithmetic  is  studied  for  mental  discipline  or  for  practical 
uiustury  over  the  every-day  problems  of  common  life,  mechanical  pro- 
cesses and  routine  methods  are  of  no  value.  Pupils  can  be  trained 
to  logical  habits  of  mind  and  stimulated  to  a  high  degree  of  intel- 
lectual energy  by  solving  problems  adapted  to  their  capacities. 
They  become  practical  arithmeticians,  not  by  learning  special 
business  forms,  but  by  founding  their  knowledge  on  reasoning 
which  they  fully  comprehend,  and  by  being  so  thoroughly  exer- 
cised in  logical  analysis  that  they  are  independent  of  arbitrary 
rules. 

The  book  contains  a  great  number  of  well-graded  and  progres- 
sive problems,  made  up  for  youths  from  ten  to  fourteen  years  of 
age.  Definitions  and  explanations  are  made  as  brief  and  simple 
as  possible.  It  is  not  intended  that  definitions  shall  be  committed 
to  memory,  but  that  they  shall  be  simply  discussed  by  teacher 
and  pupils.  Every  teacher,  of  course,  will  be  at  liberty  to  give 
better  definitions,  and  to  make  a  better  presentation  of  methods, 
than  those  given  in  the  book.  In  short,  the  chief  object  in  view 
will  be  gained  if  pupils  are  trained  to  solve  the  problems  by  neat 
and  intelligent  methods,  and  are  kept  free  from  set  rules  and 
formulas. 

A  great  many  number- problems  are  given  in  the  first  pages  of 
the  book,  so  that  the  necessary  facility  and  accuracy  in  computing 


£1306072 


IV  PREFACE. 


under  the  four  fundamental  rules  may  be  acquired  ;  as  want  of 
accuracy  and  rapidity  in  mere  calculations  distracts  the  attention 
which  should  be  given  to  the  investigation  and  correct  statement 
of  clothed  exercises.  The  pupil  should  be  required  to  do  only  so 
many  of  these  number-problems  as  are  found  to  be  necessary  to 
give  him  facility  and  accuracy  in  the  four  fundamental  operations , 
and  he  should  be  allowed  to  omit  some  of  the  harder-clothed  prob- 
lems until  he  reviews  the  book. 

The  chapter  on  the  Metric  System  is  put  at  the  end  of  the  l»o«»k 
because  many  grammar-school  pupils  have  no  time  for  it,  while 
those  who  have  time  can  as  well  learn  the  system  at  this  stage  of 
their  progress  as  earlier. 

The  chapter  on  Miscellaneous  Problems  is  intended  as  a  review 
of  the  subject-matter  of  Arithmetic  and  as  a  test  of  the  learner's 
knowledge. 

The  author  is  under  obligations  to  many  teachers  who  have 
given  valuable  suggestions  and  assistance  in  the  preparation  ot 
this  work. 

G.   A.   WENTWORTH, 
PHILLIPS  EXETER  ACADEMY,  April,  1889. 


CONTENTS. 


PAGE 

CHAPTER   I.  NOTATION  AND  NUMERATION  ....  1 

II.  ADDITION 12 

III.  SUBTRACTION 31 

IV.  MULTIPLICATION  ......  46 

V.  DIVISION 59 

VI.  DECIMALS 76 

VII.  MULTIPLES  AND  MEASURES     ....  98 

VIII.  COMMON  FRACTIONS Ill 

IX.  COMPOUND  QUANTITIES 155 

X.  PKRCENTACIK 203 

XI.  INTEREST  AND  DISCOUNT 225 

XII.  PROPORTION 254 

XIII.  POWERS  AND  ROOTS 266 

XIV.  MENSURATION 278 

XV.  MISCELLANEOUS  PROBLEMS               .        .        .  293 

XVI.  METRIC  SYSTEM   .  319 


VOCABULARY. 


Abstract  number.  This  phrase  is  employed  to  designate  numbers 
used  without  reference  to  any  particular  unit,  as  8,  10,  21.  But 
all  numbers  are  in  themselves  abstract  whether  the  kind  of  thing 
numbered  is  or  is  not  mentioned. 

Addition.  The  process  of  combining  two  or  more  numbers  so  as  to 
form  a  single  number. 

Aliquot  part.  A  number  which  is  contained  an  integral  number  of 
times  in  a  given  number.  Thus,  5,  6},  12J,  16J,  are  aliquot  parts 
of  100. 

Amount.  The  sum  of  two  or  more  numbers.  In  Interest,  the  sum 
of  principal  and  interest. 

Analysis.  The  separation  of  a  question  into  parts,  to  be  examined 
each  by  itself. 

Antecedent.     The  first  of  the  two  terms  named  in  a  ratio. 

Area  of  a  surface.  The  area  of  a  surface  is  the  number  of  units  of 
surface  it  contains  ;  the  unit  of  surface  being  a  square  whose  side 
is  a  unit  of  length. 

Arithmetic.  The  science  that  treats  of  numbers  and  the  methods 
of  using  them. 

Assets.  All  the  property  belonging  to  an  estate,  individual,  or  cor- 
poration. 

Average.  The  mean  of  several  unequal  numbers,  so  that,  if  substi- 
tuted for  each,  the  aggregate  would  be  the  same. 

Bank.  An  establishment  for  the  custody,  loaning,  and  exchange  of 
money ;  and  often  for  the  issue  of  money. 

Bank  discount.  An  allowance  received  by  a  bank  for  the  loan  of 
money,  paid  at  the  time  of  lending  as  interest. 

Bonds.  Written  contracts  under  seal  to  pay  specified  sums  of  money 
at  specified  times,  issued  by  national  governments,  states,  cities, 
and  other  corporations. 

Cancellation.  The  striking  out  of  a  common  factor  from  the  divi- 
dend and  divisor. 


viii  VOCABULARY. 

Commission.  Compensation  for  the  transaction  of  business,  reck- 
oned at  some  per  cent  of  the  money  employed  in  the  transaction. 

Common  denominator.  A  denominator  common  to  two  or  more 
fractions. 

Common  factor.     A  factor  common  to  two  or  more  numbers. 

Common  multiple.     A  multiple  common  to  two  or  more  numbers. 

Complex  fraction.  A  fraction  that  has  a  fraction  in  one  or  both  of 
its  terms. 

Composite  number.  The  product  of  two  or  more  integral  factors, 
each  factor  being  greater  than  unity. 

Compound  denominations.  Several  denominations  used  to  express 
parts  of  one  quantity. 

Compound  interest.  When  the  interest  due  is  left  unpaid,  and  con- 
sidered as  an  increase  made  to  the  principal,  the  whole  interest, 
accruing  in  any  time,  is  called  compound  interest. 

Compound  fraction.     A  fraction  of  another  fraction. 

Concrete  number.  A  phrase  used  to  denote  numbers  applied  to 
specified  things  ;  as  6  horses,  8  desks. 

Consequent.     The  second  of  the  two  terms  named  in  a  ratio. 

Consignee.     The  person  or  firm  to  whom  goods  are  sent. 

Consignor.     The  person  or  firm  who  sends  goods  to  another. 

Corporation.  An  association  of  individuals  authorized  by  law  to 
transact  business  as  a  single  person. 

Couplet.     The  two  terms  of  a  ratio  taken  together. 

Coupon.  A  certificate  of  interest  attached  to  a  bond,  to  be  cut  off 
when  due  and  presented  for  payment. 

Creditor.     A  person  or  firm  to  whom  money  is  due. 

Cube  root.     One  of  the  three  equal  factors  of  a  number. 

Customs.  Duties  or  taxes  imposed  by  law  on  merchandise  imported, 
and  sometimes  on  merchandise  exported. 

Debtor.     A  person  who  owes  money  to  another. 

Decimal  fractions.  Fractions  of  which  only  the  numerators  are 
written,  and  the  denominators  are  ten  or  some  power  of  ten. 

Decimal  point.     A  dot  placed  after  the  units'  figure  to  mark  its  place. 

Decimal  system.  The  common  system  of  numbers  founded  on  their 
relations  to  ten,  ten  tens,  etc. 

Denominator.  The  number  which  shows  into  how  many  equal  parts 
a  unit  is  divided. 

Difference.  The  number  which,  added  to  a  given  number,  makes  a 
sum  equal  to  another  given  number. 


VOCABULARY.  ix 

Discount.  Allowance  made  for  the  payment  of  money  before  it  be- 
comes due.  Also,  the  difference  between  the  market  value  and 
the  face  value  when  the  market  value  is  below  the  face  value. 

Dividend.  In  division,  the  given  number  which  is  equal  to  the 
product  of  a  given  factor  (called  divisor)  and  required  factor 
(called  quotient).  In  business,  the  share  of  profits  which  belongs 
to  each  owner  of  stock,  on  his  proportion  of  the  capital. 

Division.  The  operation  by  which,  when  a  product  and  one  of  its 
factors  are  given,  the  other  factor  is  found. 

Divisor.     The  number  by  which  a  given  dividend  is  to  be  divided. 

Draft.  A  written  order  directing  one  person  to  pay  a  specified  sum 
of  money  to  another. 

Drawee  of  a  draft.  The  person  to  whose  order  the  sum  of  money 
named  in  a  draft  is  to  be  paid. 

Drawer  of  a  draft.     The  person  who  signs  the  draft. 

Duties.  Taxes  required  by  the  government  to  be  paid  on  goods 
imported,  exported,  or  put  on  the  market  for  consumption. 

Equation.     A  statement  that  two  expressions  of  number  are  equal. 

Equation  of  payments.  The  finding  of  an  average  time  at  which 
several  payments  may  be  justly  made. 

Exchange.  A  system  of  paying  debts,  due  to  persons  living  at  a 
distance,  by  transmitting  drafts  instead  of  money. 

Exponent.  A  small  figure  placed  at  the  right  of  a  number  to  show 
how  many  times  the  number  is  taken  as  a  factor. 

Extremes.     The  first  and  last  terms  of  a  proportion. 

Evolution.     The  process  of  finding  the  root  of  a  number. 

Factors.  The  factors  of  a  number  are  a  set  of  numbers  whose  prod- 
uct is  the  given  number ;  they  are  assumed  to  be  integral,  except 
in  the  extraction  of  roots.  In  commerce,  agents  employed  by 
merchants  to  transact  business. 

Figures.  Symbols  used  to  represent  numbers  in  the  common  system 
of  notation.  Also  diagrams  used  to  represent  geometrical  forms. 

Firm.     The  name  under  which  a  company  transact  business. 

Fractions.  One  or  more  of  the  equal  parts  into  which  the  unit  is 
divided. 

Grace.  An  allowance  of  three  days,  after  the  date  a  note  becomes 
due,  within  which  to  pay  the  note. 

Gram.     The  unit  of  weight  in  the  metric  system. 

Greatest  common  measure.  The  greatest  number  which  is  a  com- 
mon factor  of  two  or  more  given  numbers. 


x  VOCABULARY. 

Improper  fraction.  A  fraction  whose  numerator  equals  or  exceeds 
'.ne  denominator. 

Index.  A  figure  written  at  the  left  and  above  the  radical  sign  to 
show  what  root  of  the  number  under  the  radical  sign  is  required. 
A  fraction  written  at  the  right  of  a  number,  of  which  the  nume- 
rator shows  the  required  power  of  that  number,  and  the  denomi- 
nator the  required  root  of  that  power. 

Instalment.     A  payment  in  part. 

Insurance.  A  guarantee  of  a  specified  sum  of  money  in  the  event 
of  loss  of  property  by  fire,  storm  at  sea,  or  other  disaster;  or  of 
loss  of  life. 

Integral  number.     A  number  which  denotes  whole  things. 

Interest.     Money  paid  for  the  use  of  inoiiey. 

Involution.     The  process  of  finding  a  power  of  a  number. 

Latitude  of  a  point.  The  angle  made  by  ilia  vertical  line  at  that 
point  with  the  plane  of  the  equator. 

Least  common  multiple.  The  least  number  which  is  a  common 
multiple  of  several  given  numbers. 

Liability.     A  debt,  or  obligation  to  pay. 

Line.  Length  without  breadth  or  thickness.  The  path  of  a  moving 
point. 

Liter.  The  unit  of  capacity  in  the  metric  system  equal  in  volume 
to  a  cube  each  edge  of  which  is  one- tenth  of  a  meter. 

Long  division.  The  method  of  dividing  in  which  the  processes  are 
written  in  full. 

Longitude  of  a  point.  The  angle  between  two  planes  supposed  to 
pass  through  the  centre  of  the  earth  and  to  contain,  the  one  the 
meridian  of  that  point,  and  the  other  the  standard  meridian. 

Loss.     The  excess  of  the  cost  price  above  the  selling  price. 

Maturity  of  a  note.     The  date  at  which  a  note  legally  becomes  due. 

Mean  proportional.  A  number  which  is  both  the  second  and  third 
terms  of  a  proportion. 

Means.     The  terms  of  a  proportion  between  the  extremes. 

Meter.     The  unit  of  length  in  the  metric  system. 

Minuend.  The  given  number  in  subtraction  which  is  equal  to  the 
sum  of  another  given  number  called  the  subtrahend,  and  a 
required  number  called  the  difference  or  remainder. 

Mixed  number.  A  number  that  expresses  both  entire  things  and 
parts  of  things  taken  together. 


VOCABULARY.  xi 

Multiple  of  a  number.     The  product  obtained  by  taking  the  given 

number  an  integral  number  of  times. 
Multiplicand,     The  number  to  be  multiplied  by  another. 
Multiplication.     The  operation  of  finding  a  number  bearing  the 

same  ratio  to  the  multiplicand  which  the  multiplier  bears  to  unity 
Multiplier.     The  number  by  which  the  multiplicand  is  multiplied. 
Net  proceeds.     The  money  that  remains  of  the  money  received  for 

property  after  all  expenses  and  discounts  are  paid. 
Notation.     A  system  of  expressing  numbers  by  symbols. 
Note.     A  written  agreement  to  pay  a  specified  sum  of  money  at  a 

specified  time. 

Number.     The  answer  to  the  question,  How  many  ? 
Numeration.    A  system  of  naming  numbers. 
Obligation.    A  debt,  or  liability  to  pay. 
Order  of  units.     A  name  used  to  designate  the  number  of  things 

in  a  group,  as  tens,  hundreds,  thousands,  etc. 
Partial  payment.     Part  payment  on  a  note. 
Partnership,     An  association  of  two  or  more  persons  to  carry  on 

business. 

Par  value.     Face  or  nominal  value. 
Pendulum.     A  body  suspended  by  a  straight  line  from  a  fixed  point, 

and  moving  freely  about  that  point  as  a  centre. 
Percentage,     A  part  of  any  given  number  reckoned  at  some  rate 

per  cent. 

Period.     A  group  of  three  figures. 
Policy     A  written  contract  of  insurance. 
Poll  tax.     A  tax  levied  by  the  head  or  poll. 
Power.     The  product  of  two  or  more  equal  factors. 
Premium.     Money  paid  for  insurance  computed  at  some  rate  per 

cent  of  the  value   insured      Also   the   excess   of  market  value 

above  par  value. 

Present  worth.     The  present  value  of  a  debt  due  at  some  future  day. 
Prime  number.    A  number  which  has  no  integral  factors  except 

itself  and  one. 

Principal.     Money  drawing  interest 
Problem.     A  question  to  be  solved, 
Product.    The  result  obtained  by  multiplying  the  multiplicand  by 

the  multiplier 
Profit.     The  excess  of  selling  price  aoove  cost 


xii  VOCABULARY. 

Proof  The  evidence  by  which  the  accuracy  of  any  result  is  estab- 
lished. 

Proper  fractioi  .  A  fraction,  the  numerator  of  which  is  less  than 
the  denominator. 

Proportion.     A  statement  that  two  ratios  are  equal. 

Quantity.     The  answer  to  the  question,  How  much  ? 

Quotient.     The  number  sought  in  division. 

Rate  per  cent.     Rate  by  the  hundred. 

Ratio.     The  relative  magnitude  of  two  numbers  or  of  two  quantities. 

Reciprocal  of  a  number.     One  divided  by  that  number. 

Reduction.  The  process  of  changing  the  unit  in  which  a  quantity 
is  expressed  without  changing  the  value  of  the  quantity. 

Remainder.  The  number  which,  added  to  the  iubtrahend,  gives  a 
sum  equal  to  the  minuend. 

Root  of  a  number.     One  of  the  equal  factors  of  the  number. 

Rule.     The  statement  of  a  prescribed  method. 

Security.     Property  used  to  guarantee  the  payment  of  any  debt. 

Share.  One  of  a  certain  number  of  equal  parts  into  which  the  capi- 
tal of  a  company  is  divided. 

Short  division.  The  method  of  dividing  in  which  the  operations  of 
multiplying  and  subtracting  are  performed  mentally. 

Solid.     A  magnitude  which  has  length,  breadth,  and  thickness. 

Solution.    The  process  by  which  the  answer  to  a  question  is  obtained. 

Specific  gravity  of  a  substance.  The  ratio  of  the  weight  of  a 
given  volume  of  it  to  that  of  an  equal  volume  of  water. 

Square  root.     One  of  two  equal  factors. 

Stock.     Capital  invested  in  business. 

Subtraction.  The  process  of  finding  a  number  which  added  to  one 
of  two  given  numbers  will  produce  the  other. 

Sum.  The  number  which  results  from  combining  two  or  more  num- 
bers by  addition. 

Surd.  An  indicated  root  the  value  of  which  cannot  be  exactly  ex- 
pressed in  figures. 

Surface.     That  which  has  only  length  and  breadth. 

Thermometer.     An  instrument  for  measuring  heat. 

Units.  The  standards  by  whicK  we  count  separate  objects  or  measure 
magnitudes. 

Verify.     To  establish,  by  trial,  the  truth  of  any  statement. 

Volume  of  a  solid.  The  volume  of  a  solid  is  the  number  of  units  of 
volume  it  contains ;  the  unit  of  volume  being  a,  cube  whose  edge 
is  a  unit  of  length. 


GRAMMAR  SCHOOL  ARITHMETIC. 

• 

CHAPTER   I. 
PRELIMINARY   DEFINITIONS. 

1,  Number,     A  fundamental  idea,  like  that  of  number, 
cannot  be  defined.     A  simple,  direct  answer  to  the  ques- 
tion "  How  many?  "  is  a  number. 

2,  A  collection  of  several  similar  objects  (as  a  collection 
of  apples)  gives  the  idea  of  number. 

3,  Units,     In  counting  separate  objects  the  standards  by 
which  we  count  are  called  units.     Thus  : 

In  counting  the  eggs  in  a  nest,  the  unit  is  an  egg. 
In  selling  eggs  by  the  dozen,  the  unit  is  a  dozen  eggs. 
In  selling  bricks  by  the  thousand,  the  unit  is  a  thousand 
bricks. 

4,  Measurement,    Continuous  magnitudes,  such  as  length, 
surface,  space,  time,  heat,   cannot   be   counted ;    they  are 
measured.    Magnitudes,  whether  continuous  or  separate,  are 
generally  measured,  and  the  standards  by  which  they  are 
measured  are  fixed  by  law,  or  by  common  consent. 

5,  Units  of  Measure.     The  standards  by  which  we  meas- 
ure magnitudes  are  called  units  of  measure.     Thus  : 

An  inch,  a  foot,  a  yard,  a  rod,  a  mile,  are  units  of  length. 
A  square  inch,  a  square  foot,  are  units  of  surface. 
A  cubic  inch,  a  cubic  foot,  are  units  of  volume. 


NOTATION   AND    NUMERATION. 


6,  Abstract  Numbers,     Numbers  standing  alone,  as  4,  7, 
13,  which  mean  4  units,  7  units,  13  units,  but  do  not  specify 
the  kind  of  objects  counted  or  the  kind  of  units  of  measure 
taken,  are  called  abstract  numbers.     They  signify  simply 
the  number  of  repetitions  of  some  unit. 

7,  Concrete  Numbers,     Expressions  that  give  the  name  of 
the  objects  counted  or  of  the  unit  of  measure  employed, 
and  the  number  of  such  objects,  or  of  such  units  of  measure, 
are   called   concrete  numbers.     Thus,   5   horses,   7   feet,   6 
pounds,  5  days,  are  called  concrete  numbers.    Such  expres- 
sions consist  of  two  parts,  the  number  proper,  and  the  kind 
of  units   taken,  and  should,  strictly  speaking,  be    called 
quantities. 

8,  Arithmetic  treats  of  the   simple   properties  of  num- 
bers, and  the  art  of  computing  by  numbers. 

NOTATION  AND  NUMERATION. 

9,  The  first  numbers  have  special  names,  as  follows : 
one,  two,  three,  four,  five,  six,  seven,  eight,  nine,  ten. 

10,  The  first  nine  of  these  numbers  are  called  Simple 
Units,  or  units  of  the  first  order. 

11,  The  group  of  ten  units  has  received  the  name  of  a 
Ten,  or  a  unit  of  the  second  order ;  and  we  count  by  tens  as 
by  units  ;  thus  : 

one  ten,  two  tens,  three  tens  ...  nine  tens,  ten  tens. 

12,  The  group  of  ten  tens  has  received  the  name  of  a 
Hundred,  or  a  unit  of  the  third  order ;   and  we  count  by 
hundreds,  as  by  tens  and  units;  thus: 

one  hundred,  two  hundreds  ...  ten  hundreds. 


NOTATION   AND   NUMERATION. 


13,  A  group  of  ten  hundreds  is  called  a  Thousand,  or  a 
unit  of  the  fourth  order. 

14,  From  ten  units  of  the  fourth  order  is   formed   a  ten 
thousand,  or  a  unit  of  the  fifth  order;  and  from  ten  units  of 
the  fifth  order  is  formed  a  hundred  thousand,  or  a  unit  of 
the  sixth  order. 

15,  Units  of  the  seventh  order  are  called  Millions ;  of  the 
eighth   order,    ten  millions;    of  the   ninth  order,  hundred 
millions.      Finally,  units    of    the   tenth   order   are    called 
Billions;  units  of  the  thirteenth  order,  Trillions;  and  so  on. 

16,  The  table  of  units  of  different  orders  is  as  follows : 

First  order,  simple  units,  \ 

Second  order,         tens  of  units,  >  first  class. 

Third  order,  hundreds  of  units,  ) 

Fourth  order,         thousands,  \ 

Fifth  order,  tens  of  thousands,  >  second  class. 

Sixth  order,  hundreds  of  thousands,  J 

Seventh  order,       millions,  \ 

Eighth  order,         tens  of  millions,  >  third  class. 

Ninth  order,  hundreds  of  millions,  ) 

Tenth  order,  billions,  -\ 

Eleventh  order,     tens  of  billions,  >  fourth  class. 

Twelfth  order,  •  hundreds  of  billions,  J 

Thirteenth  order,  trillions,  \ 

I  fifth  class. 

17,  The  group  of  the  first  three  orders  is  called  the  first 
class  of  units,  and  the  group  of  the  three  following  orders, 
the  second  class,  and  so  on. 

18,  The  unit  of  the  second  class  is  equal  to  a  thousand 
units  of  the  first  class,  and  a  unit  of  the  third  class  is  equal 
to  a  thousand  units  of  the  second  class,  and  so  on. 


NOTATION   AND   NUMERATION. 


19,  To  read  a  number  we  decompose  it  into  units  of  the 
different  orders,  and  state  how  many  groups  there  are  of 
each  kind,  commencing  with  the  highest  order.     Thus,  for 
example,    two    millions,    three    thousands,    five    hundreds, 
seven  tens,  and  four  units. 

20,  It  is  clear  that  the  names  of  all  numbers  up  to  a 
billion  are  formed  by  combining  the  names  of  the  first  nine 
numbers  with  the  words  ten,  hundred,  thousand,  million. 

21,  Usage  sanctions  the  following  irregularities  : 

I.  Instead  of  saying  two  tens,  three  tens,  four  tens,  five 
tens,  six  tens,  seven  tens,  eight  tens,  nine   tens,  we   say 
twenty,  thirty,  forty,  fifty,  sixty,  seventy,  eighty,  ninety. 

II.  The  names  of  the  numbers  between  ten  and  twenty 
are    eleven,    twelve,    thirteen,    fourteen,    fifteen,    sixteen, 
seventeen,  eighteen,  nineteen. 

22,  The  names  of  the  numbers  between  twenty  and  a 
hundred  are : 

twenty-one,  twenty-two,  twenty-three  ...  twenty-nine, 
thirty-one,  thirty-two,  thirty-three  ...  thirty-nine, 

ninety-one,  ninety-two,  ninety-three  ...  ninety-nine. 

• 

23,  The  names  of  the  numbers  between  a  hundred  and 
a  thousand  are : 

hundred  one,  hundred  two  ...  hundred  ninety-nine, 
two  hundred  one  ...  two  hundred  ninety-nine, 

nine  hundred  one  ...  nine  hundred  ninety-nine. 

24,  The  common  system  of  notation  employs  ten  figures 
or  digits  : 

1,   2,   3,   4,   5,   6,    7,   8,   9,   0. 


NOTATION   AND    NUMERATION. 


The  first  nine  of  these  figures  represent  the  first  nine  num- 
bers ;  the  last,  which  is  called  Zero,  Naught,  or  Cipher,  is 
used  to  denote  the  absence  of  units  of  the  order  in  which 
it  stands.  It  is  possible  to  express  all  numbers  by  these 
ten  digits  by  making  the  value  of  each  figure  increase  ten- 
fold for  every  place  that  it  is  moved  to  the  left. 

25,  If  we  have  given  a  number  written  in  figures,  the 
position  of  each  figure  counting  from  the  right  indicates 
the  order  of  units  that  the  figure  represents.  If  we  divide 
the  number  into  periods  of  three  figures  each,  the  first 
period  on  the  right  will  be  the  period  of  simple  units,  the 
second  period  will  be  the  period  of  thousands,  the  third 
will  be  the  period  of  millions,  and  so  on.  In  each  period 
the  first  figure  on  the  right  expresses  the  units  of  that 
class,  the  second  figure  the  tens,  and  the  third  the 
hundreds.  Thus : 

MILLIONS.  THOUSANDS.  UNITS. 


Tens.          Units.        Hundreds.        Tens,  Units.          Hundreds.        Tens.  Units. 

21  334  334 

Thus,  the  number  21,334,334  means  and  is  read  21 
millions,  334  thousands,  334  units.  If  the  number  is 
applied  to  dollars,  it  means  and  is  read  21  million,  334 
thousand,  334  dollars.  The  next  period  is  the  billions' 
period. 

NOTE.  The  fundamental  principle  of  forming  and  expressing 
numbers  should  be  illustrated  by  making  little  bundles  of  wooden 
toothpicks,  ten  in  each  bundle,  and  then  making  bundles  of  hun- 
dreds by  taking  for  each  hundred  ten  bundles  of  ten  each.  When 
the  pupil  has  become  familiar  with  forming  and  expressing  numbers 
consisting  of  hundreds,  tens,  and  units,  he  should  be  shown  that  the 
method  of  forming  and  expressing  numbers  of  hundreds,  tens,  and 
units  of  thousands  is  precisely  the  same,  the  only  difference  being 
that  the  unit  of  this  period  is  not  a  single  toothpick,  but  a  pile  of  ten 
bundles  of  a  hundred  each,  which  is  a  thousand. 


6  NOTATION    AND   NUMERATION. 

26,  To  write  a  number  in  figures  we  write  successively  trie 
number  of  units  of  each  order  from  left  to  right,  beginning 
at  the  highest  order  and  taking  care  to  supply  by  zeros 
orders  of  units  that  may  be  lacking. 

27,  To  read  a  number  written  in  figures  we  divide  the 
number  into  periods  of  three  figures  each  from  right  to 
left :  this  done,  we  begin  to  read  at  the  left-hand  period 
and  read  as  if  the  figures  of  that  period  stood  alone,  adding 
the  name  of  the  period ;  then  the  next  period  to  the  right 
is  read  with  the  name  of  that  period,  and  so  on. 

28,  The   number   1256  may  be  read  one  thousand  two 
hundred  fifty-six,  or  it  may  be  read  twelve  hundred  fifty-six. 
The  number  5004  may  be  read  five  thousand  four,  or  it 
may  be  re&d  fifty  hundred  four.      The  shortest  method  is  the 
best  method  of  reading  any  number.    Twelve  hundred  fifty- 
six  is  shorter  than  one  thousand  two  hundred  fifty-six ;  five 
thousand  four  is  shorter  than  fifty  hundred  four. 

29,  It  will  be  seen  that  the  value  of  each  figure,  in  any 
number  expressed  in  figures,  depends  on  two  things : 

First,  the  value  attached  to  the  figure  without  regard  to 
its  position. 

And,  secondly,  the  value  it  acquires  from  the  place  it 
holds  in  the  number. 

The  value  of  a  figure,  without  regard  to  its  position,  is 
called  its  absolute  value ;  and  the  value  it  acquires  by  its 
position  is  called  its  local  value. 

30,  The  art  of  expressing  numbers  by  means  of  figures 
is  called  Notation,  and  the  art  of   expressing   in  words  a 
number  written  in  figures  is  called  Numeration, 

31,  The  unit  of  money  is  the  dollar.     Instead  of  writing 
the  word  dollars,  this  mark  $  is  used,  which  is  called  the 


NOTATION   AND    NUMERATION. 


sign  for  dollars,  or  the  "  dollar  mark."  Thus,  if  we  wish 
to  write  five  dollars,  we  write  it  $5. 

It  takes  ten  ten-cent  pieces  to  make  a  dollar ;  that  is,  a 
ten-cent  piece  is  one-tenth  of  a  dollar.  It  takes  ten  single 
cents  to  be  equal  in  value  to  a  ten-cent  piece.  If  we  have 
one  dollar  and  one  ten-cent  piece,  we  write  it  $1.10.  If 
we  have  one  dollar,  one  ten-cent  piece,  and  two  cents,  we 
write  it  $1.12. 

The  dot  which  is  placed  after  the  one  dollar  is  called 
the  Decimal  Point.  Figures  to  the  left  of  the  decimal  point 
denote  whole  units.  Figures  to  the  right  of  the  decimal 
point  denote  parts  of  a  unit,  and  are  called  Decimal  Frac- 
tions, The  expression  $1.10  is  read  "one  dollar  and  ten 
cents"  ;  and  the  expression  $  1.12  is  read  "one  dollar  and 

twelve  cents." 

Ex.  ±. 
Write  in  figures : 

1.  Two  hundred  thirty-six,  one  hundred  forty,  five  hun- 

dred two,  seven  hundred  three. 

2.  Five  hundred  fourteen,  three   hundred  seventy-six, 

four  hundred  thirty,  eight  hundred  two,  nine  hun- 
dred twenty-seven. 

3.  One  hundred  ninety,  four  hundred  six,  eight  hundred 

ten,  two  hundred  seven. 

4.  Three  hundred  ten,  two  hundred  thirteen,  six  hun- 

dred twenty-three,  two  hundred  nineteen. 

5.  Five  hundred  fifty,  four  hundred  four,  four  hundred 

twenty-five,  eight  hundred  sixty. 

6.  Eight   hundred   sixteen,  seven  hundred  eight,  nine 

hundred,  seven  hundred  three. 

7.  Nine  hundred  ninety-five,  eight  hundred  eighty,  seven 

hundred,  eignt  hundred  seven. 

8.  Two  hundred  seventeen,  four  hundred  twelve,  four 

hundred  eight,  one  hundred  two. 


8  NOTATION   AND   NUMERATION. 

9.    Four  hundred  seventeen,  six  hundred  nineteen,  three 
hundred  six,  one  hundred  eighteen. 

Ex.  2. 
Read  (or  write  in  words)  : 

1.  500,   700,   300,  200,  900,  100. 

2.  830,   709,   506,  350,  819,  703. 

3.  607,   312,   918,  810,  103,  560. 

4.  752,   698,   405,  536,  121,  514. 

5.  973,   356,   703,  409,  211,  713. 

6.  225,     64,   970,  49,  83,  674. 

7.  106,   170,   380,  759,  921,  538. 

8.  481,   360,   593,  32,  296,  551. 

9.  182,   802,   555,  705,  '649,  630. 
10.  314,     97,   613,  384,  992,  516. 

Ex.  3. 

Write  in  figures : 

1.  Eight  thousand  seven  hundred  three,  four  thousand 

forty-five,  six  thousand  three  hundred  eight,  forty- 
eight  hundred. 

2.  Five  thousand  forty-eight,  nineteen  hundred  ninety, 

seven  thousand  eighty-two,  eight  thousand  fifty. 

3.  Seven  thousand   two  hundred  forty,  nine   thousand 

nine  hundred  nineteen,  six  thousand  seven,  eight 
thousand  seven  hundred  seventy-six. 

4.  Seven   thousand   one   hundred   seven,    six   thousand 

eight  hundred  four,  nine  thousand  one  hundred  ten, 
five  thousand  five  hundred  fifty. 

5.  Six   thousand   eighty-six,    four   thousand   forty,    one 

thousand  ten,  nine  thousand  ninety-nine. 

6.  Eight  thousand  eighty,  seventeen  hundred  fifty-seven, 

eleven  hundred  one,  seven  thousand  seven,  forty- 
five  hundred  forty-five 


NOTATION   AND   NUMERATION. 


7.  Two  thousand  four  hundred  ninety-six,  eighteen  hun- 

dred eighty-three,  three  thousand  ninety-five,  one 
thousand  eleven. 

8.  One  thousand   thirteen,  one  thousand  one,  fourteen 

hundred,  thirty-three  thousand  fourteen. 

9.  Seventeen  hundred  thirty-six,  three  thousand  forty- 

nine,  eight  thousand  eighteen,  nine  thousand  seventy. 
10.    Four  thousand  seven  hundred  nine,  fifteen  hundred  ten, 
one  thousand  sixty-nine,  sixteen  thousand  sixteen. 


Ex.  4. 

Read  (or  write  in  words)  : 

1. 

8,000, 

5,000, 

2,000, 

6,000, 

1,000, 

9,000. 

2. 

9,210, 

6,907, 

7,402, 

9,998, 

4,060, 

7,210. 

3. 

5,068, 

4,020, 

1,400, 

7,031, 

1,290, 

1,010. 

4. 

8,808, 

6,006, 

8,482, 

3,096, 

4,720, 

11,973. 

5. 

12,002; 

11101 

5,812, 

1,739, 

6760, 

6,903. 

6. 

4,085, 

1,169, 

2,615 

5,007, 

1,110, 

1,460. 

7. 

4,760, 

4,190, 

2,607, 

5,180, 

1,200, 

3,746. 

8. 

9,008, 

8,300, 

6,804, 

2,977, 

6,202, 

9,620. 

9. 

6,322, 

7,450, 

8,673, 

2,603, 

2,518, 

1,508. 

10. 

7,080, 

1,009, 

8,070, 

5,068, 

1,397, 

5,782. 

Write  in  figures :  Ex*  5' 

1.  Twelve  and  twelve  hundredths,  twenty-two  and  eight 

tenths,  three  hundred  twenty-five  and  six  tenths, 
one  hundred  one  and  one  hundred  one  thousandths. 

2.  Seventy-five   and    seventy-five    hundredths,    eighty- 

three  and  twenty-six  thousandths,  ninety-six  and 
seven  hundred  four  thousandths,  one  thousand  ten 
and  two  tenths. 

3.  Five  hundred  seventy-three  and  five  hundred  seventy- 

three  thousandths,  eleven  thousand  four  and  sixteen 
hundredths,  three  hundred  sixty-five  and  eighl 
tenths,  seventy-two  and  ninety-six  hundredths. 


10 


NOTATION  AND   NUMERATION. 


4.  Three  and  nineteen  thousandths,   six  hundred  fifty- 

eight  and  two  hundredths,  eight  hundred  and  eight 
hundredths,  thirty-seven  and  five  thousandths. 

5.  Seventy-one  and  seven  tenths,  seven  and  seventeen 

hundredths,  seven  hundred  and  seventeen  thou- 
sandths, eight  hundred  ten  and  one  tenth. 

6.  Eighty-one  and  one  hundredth,  eight  and  one  hundred 

one  thousandths,  nine  hundred  sixty-three  and  two 
tenths,  ninety-six  and  thirty-two  hundredths,  nine 
and  six  hundred  thirty-two  thousandths. 

7.  Six  hundred  and  five  tenths,  sixty  and  five  hundredths, 

six  and  five  thousandths. 

8.  Nine  hundred  eighty-three  and  three  tenths,  ninety- 

eight  and  thirty-three  hundredths,  nine  and  eight 
hundred  thirty-three  thousandths. 

9.  One   hundred    twelve   and    one   tenth,    eleven    and 

twenty-one  hundredths,  one  and  one  hundred 
twenty-one  thousandths. 

10.  Eleven  thousand  and  sixty-three  thousandths,  twenty- 
three  and  eighty-six  hundredths,  one  hundred  ten 
and  eleven  hundredths. 


Ex. 
Read  (or  write  in  words) : 


6. 


1 

2. 
3. 
4. 
5. 

a 

7. 
8. 
9. 
10. 

3010.3, 
903.9, 
234.5, 
6187.8, 
291.59, 
360.4, 
47.S28, 
510.14, 
65.002, 
770.85, 

477.12, 
413.9, 
3010.3, 
785.33, 
29.645, 
3605.9, 
59.184, 
51.028, 
69.949, 
6994.9, 

60.206, 
17.918, 
59.106, 
90.849, 
30.081, 
361.16, 
600.65, 
580.35, 
602.17, 
712.06. 

698.97, 
113.94, 
43.136, 
92.294, 
299.07, 
39.041, 
601.19, 
5804.7, 
6020.6, 
719.66, 

778.15, 
14.613, 
380.21, 
27.989, 
30.190, 
468.64, 
60.108, 
641.97, 
64.058, 
833.87, 

84.510. 
204.12. 
361.73. 
28.012. 
35.257. 
463.59. 
52.466. 
6409.8. 
76.343. 
83.493. 

NOTATION   AND   NUMERATION  11 

Ex.  7. 

Write  in  figures : 

1.  Fifty  thousand  three  dollars,    eighty  thousand   nine 

hundred  ninety  dollars. 

2.  Twenty-eight  million  seven  hundred  forty-four  thou- 

sand one  hundred  sixty-nine  dollars. 

3.  Five  hundred  sixteen  dollars  and  ten  cents,  twenty- 

five  hundred  fifty  dollars  and  sixty-nine  cents. 

4.  Sixteen  hundred  million  thirty  thousand  three  hun- 

dred eight  dollars  and  fifty  cents. 

5.  Twenty-seven  hundred  million  one  thousand  one  dol- 

lars and  eighty-seven  cents. 

6.  Five  hundred  thousand  two  hundred  one  dollars  and 

seventy-five  cents. 

7.  Eight    million     fourteen    thousand     three    hundred 

twenty-five  dollars  and  twenty-five  cents. 

8.  Ninety-seven  million  two  hundred  thousand  one  hun- 

dred two  dollars  and  five  cents. 

9.  Ten  million  ten  thousand  ten  dollars  and  ten  cents. 
10.    Eleven   hundred    ten   thousand    dollars   and   eleven 

cents. 

Ex.  8. 
Bead  (or  write  in  words) : 

1.  $259,132.10,  $27,186.25. 

2.  $1,213,062.50,  $2,763,001.75. 

3.  $3,675,321.12,  $3,500,005.15. 

4.  $17,360,502.20,  $27,132,857.33. 

5.  $55,333,263.36,  '  $58,785,587.09. 

6.  $116,001,556.40,  $275,363,750.11. 

7.  $  660,878,640.69,  $  594,340,000.94. 

8.  $600,241,560.02,  $124,271,000.01.       , 

9.  $768,301,520.20,  $802,631,516.73. 
10.     $505,631,880.04,  $1,555,676,410.62. 


CHAPTER   II. 

ADDITION. 

32.  If  you  put  2  cents  with  3  cents,  how  many  cents 
have  you  ?     Answer,  5  cents. 

How  can  you  express  this  operation  on  your  slate? 

You  can  write  the  figure  2 ;  then  the  figure  3  be- 
neath it;    draw  a  line  underneath,  and   below  the 
line  write  the  figure  5.     The  work  is  shown  in  the         5 
margin. 

Or,  you  can  express  it  thus :  2  +  3  =  5. 

The  sign  +  is  called  plus,  and  means  that  the  numbers 
between  which  it  is  placed  are  to  be  counted  together ;  and 
.  the  sign  =  means  equals,  So  that  2  +  3  —  5  is  read  2  plus 
3  equals  5. 

33.  The  operation  of  finding  a  number  equal  to  two  or 
more  numbers  taken  together  is  called  addition ;   and  the 
result  is  called  their  sum,     The  numbers  to  be  added  are 
called  addends, 

Name  the  sums  of  the  following  numbers,  and  practise 
naming  them  until  you  can  name  each  sum  the  instant 
your  eye  rests  upon  the  numbers  to  be  added. 

Ex.  9.     (Oral) 

1  +  1-  3+1-  1+0-  1+7-  1+5= 
2+1=  8+1-  1+4-  6+1-  9+1= 
2+2-  2+0-  1+2-  8+2-  2+7- 


ADDITION. 

13 

2  +  5  = 

6  +  2- 

3  +  2  = 

2  +  4  = 

2  +  9  = 

3  +  4  = 

2  +  3- 

6  +  3  = 

1  +  3  = 

3  +  7  = 

8  +  3  = 

3  +  0  = 

3  +  6  = 

3  +  3  = 

9  +  3  = 

0  +  4  = 

5  +  4  = 

4  +  7  = 

9  +  4  = 

4  +  1  = 

8  +  4-= 

4  +  3  = 

2  +  4  = 

4  +  4  = 

4  +  6  = 

5  +  5  = 

5  +  7  = 

3  +  5  = 

0  +  5  = 

5  +  9- 

2  +  5  = 

5  +  1  = 

4  +  5  = 

5  +  6  = 

8  +  5- 

6  +  3  = 

1  +  6  = 

5  +  6  = 

6  +  0  = 

2  +  6- 

6  +  6  = 

4  +  6  = 

6  +  9  = 

7  +  6  = 

6  +  8  --- 

5  +  7  = 

5  +  6  = 

7-1  3  = 

7  +  1  = 

0  +  7  = 

8  +  7  = 

7  +  2  = 

4+7  = 

7  +  7  = 

7  +  9=- 

8  +  1  = 

5  +  8  = 

2  +  8  = 

8  +  0  = 

7  +  8  - 

8  +  3  = 

8  +  8  = 

4  +  8  = 

8  +  6  = 

9  +  8=- 

9  +  0  = 

9  +  9  = 

9  +  2  = 

1+9  = 

3  +  9  = 

4  +  9  = 

9  +  6  = 

8  +  9  = 

9  +  5  = 

7  +  9  = 

5        6 

8         7 

4         3 

6         6 

6         7 

3        7 

3         5 

5         6 

4         8 

5         5 

7        7 

7         7 

7         8 

5         8 

8         8 

7        2 

9         4 

8         2 

8         7 

9         8 

9        9 

9         9 

6         2 

3         8 

3         6 

1        4 

7         8 

8         5 

4         2 

6         5 

8        7 

5         3 

7         8 

9         5 

4         3 

7        2 

4        8 

3         8 

3         .3 

3         3 

14  ADDITION. 


Ex.  10. 
Find  the  sums  of  the  following  numbers : 

1.   28  9-    29  17    32          25.   47          33.   49 

57985 


2.   43  10.   56  18.   58  26.   G5  34.   G7 

68999 


3.   54  11.    78  19.    79  27.    27  35.   43 

75487 


4.   63  12.    78  20.   57  28.   27  36.   35 

89869 


5.    74  13.    32  21.    63  29.   42  37.    14 

37896 


6.    18  14.    27  22.    36  30.    12  38.    73 

99898 


7.   85  15.    37  23.    59  31.    50  39.    13 

75459 


8.   19          16.   37          24.   93          32.   89          40.  79 
89976 


AUDITION  15 


Copy  the  following,  and  fill  the  blanks  : 

42  +  9=  19  +  8=  17  +  9=  23  +  8  = 

71  +  9=  26  +  7=  35  +  8=  47  +  6  = 

85  +  8=  18+7=  17  +  3=  18  +  9  = 

29  +  6=  38  +  5=  15  +  7=  14  +  8  = 

34,  Since  10  in  any  place  is  equal  to  1  in  the  next  place 
to  the  left,  if  the  sum  of  the  digits  of  any  column  exceeds  9, 
write  the  units'  figure  of  the  sum  under  the  column  added 
and  carry  the  number  of  tens  to  the  next  column. 

Thus,  in  the  following  example  :  872 

The  sum  of  the  digits  in  the  right-hand  column  is  3.  The  99^ 

sum  of  the  digits  in  the  second  column  is  16  ;  the  6  is  writ- 
ten under  this  column  and  the  1  is  carried  to  the  third          1863 
column.     The  sum  of  the  digits  of  the  third  column,  to- 
gether with  the  1  carried  to  it,  is  18  ;  the  8  is  written  under  this  col- 
umn and  the  1  is  carried  to  the  place  of  thousands. 


Add: 

1.  497  6.   689  11.   9535  16.   56902 
735  297  9675  94876 

2.  840  7.    477  12.   5557  17.    93689 
869  335  5763  60086 

3.  997  8.   449  13.   8284  18.    59857 
289  483  7998  84556 

4.  643  9.   857  14.    8956  19.   83897 
937  816  7694  50799 

5.  958  10.   842  15.   3448  20.   59988 
294  863  4876  99939 


16  ADDITION. 


35,  Practise  the  following  additions  until  you  can  name 
the  results  as  rapidly  as  you  can  count  1,  2,  3,  4,  5,  etc. 

Ex.  12.      (Oral.) 

Add  by  twos  to  50,  beginning  0,  2,  4,  6,  8.  Add  by 
twos  to  51,  beginning  1,  3,  5,  7,  9. 

Add  by  threes  to  102,  beginning  0,  3,  6.  Add  by  threes 
to  100,  beginning  1,  4,  7.  Add  by  threes  to  101,  beginning 
2,  5,  8. 

Add  by  fours  to  100,  beginning  0,  4,  8.  Add  by  fours 
to  101,  beginning  1,  5,  9.  Add  by  fours  to  102,  beginning 
2,  6,  10.  Add  by  fours  to  103,  beginning  3,  7,  11. 

Add  by  fives  to  100,  beginning  0,  5,  10.  Add  by  fives 
to  101,  beginning  1,  6,  11.  Add  by  fives  to  102,  beginning 
2,  7,  12.  Add  by  fives  to  103,  beginning  3,  8,  13.  Add 
by  fives  to  104,  beginning  4,  9,  14. 

Add  by  sixes  to  102,  beginning  0,  6,  12.  Add  by  sixes 
to  103,  beginning  1,  7, 13.  Add  by  sixes  to  104,  beginning 
2,  8,  14.  Add  by  sixes  to  105,  beginning  3,  9,  15.  Add 
by  sixes  to  100,  beginning  4,  10,  16.  Add  by  sixes  to  101, 
beginning  5,  11,  17. 

Add  by  sevens  to  105,  beginning  0,  7,  14.  Add  by  sevens 
to  106,  beginning  1,  8,  15.  Add  by  sevens  to  100,  begin- 
ning 2,  9,  16.  Add  by  sevens  to  101,  beginning  3,  10,  17. 
Add  by  sevens  to  102,  beginning  4,  11,  18.  Add  by  sevens 
to  103,  beginning  5,  12,  19.  Add  by  sevens  to  104,  begin- 
ning 6,  13,  20. 

Add  by  eights  to  104,  beginning  0,  8,  16.  Add  by  eights 
to  105,  beginning  1,  9,  17.  Add  by  eights  to  106,  begin- 
ning 2,  10,  18.  Add  by  eights  to  107,  beginning  3,  11,  19. 
Add  by  eights  to  100,  beginning  4,  12,  20.  Add  by  eights 
to  101,  beginning  5,  13,  21.  Add  by  eights  to  102,  begin- 
ning 6,  14,  22.  Add  by  eights  to  103,  beginning  7,  15,  23. 


ADDITION.  17 


Add  by  nines  to  108,  beginning  0,  9,  18.  Add  by  nines 
to  100,  beginning  1,  10,  19.  Add  by  nines  to  101,  begin- 
ning 2,  11,  20.  Add  by  nines  to  102,  beginning  3,  12,  21. 
Add  by  nines  to  103,  beginning  4,  13,  22.  Add  by  nines 
to  104,  beginning  5,  14,  23.  Add  by  nines  to  105,  begin- 
ning 6,  15,  24.  Add  by  nines  to  106,  beginning  7,  16,  25. 
Add  by  nines  to  107,  beginning  8,  17,  26. 

36,  Practise  adding  columns  of  three  digits  until  you 
can  name  the  sum  of  any  three  digits  the  instant  you  see 
them. 

Ex.  13.     (Oral.) 
Find  the  sums  of  : 

1.  3245456743 
5123323232 
4334214327 

2.  3456578345 
4242421232 
1323230423 

3.  453543    8^6    7    8 
2243151296 
3132323549 

4.  1876795796 
3657286387 
51799949    18 

5.  7659765395 
2450669898 
3348468797 


18  ADDITION. 


6. 

3 

5 

7 

6 

8 

9 

7 

6 

9 

8 

9 

6 

2 

3 

5 

6 

5 

3 

8 

8 

7 

8 

5 

4 

4 

4 

7 

5 

8 

8 

7. 

6 

6 

7 

7 

4 

4 

3 

5 

3 

6 

2 

5 

8 

7 

9 

4 

8 

5 

7 

8 

5 

3 

9 

7 

4 

4 

7 

5 

7 

7 

8. 

8 

3 

2 

3 

5 

5 

9 

4 

2 

2 

4 

3 

2 

4 

4 

6 

3 

7 

9 

2 

4 

3 

9 

8 

8 

6 

6 

5 

8 

8 

9. 

9 

5 

7 

8 

9 

6 

7 

4 

5 

8 

9 

6 

7 

4 

5 

7 

4 

3 

4 

2 

7 

6 

5 

7 

5 

6 

8 

9 

7 

9 

10. 

3 

2 

2 

3 

6 

8 

7 

8 

9 

6 

8 

9 

2 

3 

7 

5 

2 

2 

8 

9 

7 

8 

9 

7 

4 

4 

3 

2 

7 

7 

37.    The  quickest  way  to  add  columns  of  four  or  more 
digits  is  to  train  the  eye  to  see  at  a  glance  sums 

of  20,  and  simply  add  these  sums.     If  you  add  the  8 

column  given  in  the  margin  by  single  digits,  you  say  9 

to  yourself,  ten,  thirteen,  seventeen,  twenty-two,  twenty-  6 

eight,  thirty-seven,  forty-five;   if  you  add  by  taking  5 

two  digits  at  a  time,  you  say  ten,  seventeen,  twenty-  4 

eight,  forty-five;  if  you  add  by  taking  three  digits  at  3 

a  time,  you   say  thirteen,  twenty-eight,  fo?*ty-five;   if  2 

you  add  by  20's,  you  say  twenty  (separating  5  into  3  8 
and  2),  forty-five. 


ADDITION.  19 


Ex.  14. 

Find 

the  sums  of: 

1-     5 

9 

6 

4 

5 

7 

1 

7 

8 

2 

4 

5 

4 

6 

8 

3 

4 

3 

4 

6 

8 

7 

9 

5 

7 

8 

5 

6 

3 

9 

6 

3 

7 

2 

3 

6 

3 

8 

2 

hr 
< 

4 

8 

5 

8 

4 

9 

8 

5 

7 

4 

7 

4 

8 

6 

8 

2 

6 

4 

8 

5 

3 

9 

2 

9 

5 

7 

7 

3 

2 

3 

5 

7 

6 

4 

9 

3 

9 

6 

8 

5 

2.      8 

5 

4 

4 

5 

2 

3 

5 

3 

6 

2 

3 

2 

2 

3 

6 

8 

3 

8 

4 

4 

6 

5 

5 

2 

4 

6 

8 

4 

7 

9 

8 

9 

7 

8 

5 

7 

9 

9 

3 

3 

2 

7 

6 

4 

3 

9 

5 

7 

5 

7 

8 

6 

9 

8 

5 

5 

7 

3 

8 

6 

5 

9 

4 

6 

3 

4 

6 

8 

6 

5 

3 

4 

3 

2 

7 

8 

9 

6 

7 

2 

4 

1 

8 

1 

9 

2 

4 

2 

9 

3.     7 

5 

4 

2 

4 

8 

4 

6 

9 

7 

3 

6 

4 

6 

8 

7 

9 

8 

3 

9 

5 

7 

9 

8 

6 

3 

6 

0 

8 

6 

8 

3 

6 

4 

7 

5 

7 

4 

2 

3 

2 

9 

7 

7 

4 

5 

4 

5 

8 

4 

8 

4 

9 

9 

5 

4 

3 

3 

7 

5 

6 

2 

3 

8 

4 

7 

7 

6 

3 

8 

3 

8 

7 

2 

6 

9 

6 

9 

4 

6 

4 

1 

7 

3 

2 

9 

2 

7 

6 

8 

20  ADDITION. 


4.      6 

5 

9 

8 

5 

9 

8 

7 

9 

6 

8 

8 

5 

6 

9 

6 

4 

9 

3 

8 

5 

3 

3 

7 

4 

7 

5 

6 

8 

3 

3 

7 

4 

9 

3 

5 

9 

3 

9 

5 

7 

9 

9 

5 

7 

9 

7 

4 

8 

3 

4 

4 

6 

8 

2 

8 

6 

5 

7 

6 

6 

8 

8 

4 

9 

3 

8 

8 

3 

9 

8 

3 

2 

3 

6 

9 

3 

6 

4 

5 

2 

6 

1 

2 

8 

4 

2 

1 

0 

2 

5.     8 

5 

3 

8 

6 

3 

5 

1 

3 

4 

5 

9 

9 

6 

5 

4 

5 

3 

3 

3 

5 

6 

1 

4 

7 

7 

6 

5 

3 

2 

1 

1 

1 

3 

4 

3 

1 

8 

2 

6 

1 

1 

8 

4 

9 

1 

5 

5 

9 

1 

9 

7 

1 

0 

3 

4 

1 

7 

6 

6 

7 

7 

9 

4 

9 

3 

1 

4 

2 

7 

1 

1 

3 

3 

1 

1 

1 

0 

9 

8 

6 

4 

7 

9 

2 

9 

8 

9 

8 

9 

Ex.  15. 

Find  the  sums  of : 


1. 

50 

2.  40 

3.  60 

4.  30 

5.  10 

6.  80 

20 

80 

50 

10 

70 

90 

70 

20 

80 

90 

10 

30 

60 

30 

20 

40 

90 

80 

30 

70 

50 

20 

40 

60 

90 

80 

30 

50 

70 

30 

80 

60 

40 

70 

30 

40 

10 

50 

70 

80 

20 

50 

ADDITION.  21 


7.  40 

8.  80 

9.  52 

10.  30 

11.  42 

12.  60 

21 

70 

60 

23 

40 

40 

90 

31 

70 

52 

50 

32 

50 

42 

30 

91 

80 

54 

83 

60 

81 

70 

70 

90 

70 

51 

42 

33 

34 

82 

62 

90 

50 

80 

90 

91 

13.  51 

14.  56 

15.  48 

16.  36 

17.  25 

18.  17 

46 

63 

31 

42 

52 

82 

30 

72 

45 

50 

49 

25 

25 

81 

82 

81 

38 

13 

32 

17 

19 

14 

41 

80 

47 

26 

21 

35 

57 

45 

19.  18 

20.  57 

21.  15 

22.  44 

23.  19 

24.  91 

24 

31 

8 

21 

27 

42 

91 

28 

23 

36 

48 

36 

33 

63 

70 

8 

39 

82 

64 

90 

61 

14 

7 

71 

75 

9 

55 

27 

9 

54 

37 

81 

83 

59 

87 

65 

25.  48 

26.  52 

27.  8 

28.  16 

29.  33 

30.  54 

9 

61 

43 

48 

52 

46 

17 

26 

52 

85 

27 

8 

29 

28 

67 

7 

38 

19 

83 

83 

9 

26 

41 

92 

75 

94 

17 

35 

9 

57 

21 

77 

84 

,  54 

94 

83 

22  ADDITION. 


31.  55 

32.  68   33. 

9   34. 

13   35.  48 

36.  35 

67 

5 

23 

99       6 

42 

78 

43 

25 

7      51 

57 

9 

67 

68 

85       9 

64 

4 

25 

79 

64      23 

49 

18 

14 

7 

39      88 

87 

Ex.  16. 

Add: 

1.  123 

2.  516 

3.  321 

4.  225 

5.  871 

205 

341 

75 

716 

215 

310 

236 

184 

348 

64 

79 

110 

769 

519 

371 

118 

196 

815 

96 

296 

6.  123 

7.  205 

8.  310 

9.  79 

10.  118 

516 

341 

236 

110 

196 

321 

75 

184 

769 

815 

225 

716 

348 

519 

96 

871 

215 

64 

371 

296 

11.  213 

12.  421 

13.  85 

14.  231 

15.  526 

327 

87 

222 

624 

448 

98 

116 

376 

785 

379 

716 

615 

584 

923 

87 

825 

399 

972 

84 

999 

ia  213 

17.  327 

18.  98 

19.  716 

20.  825 

421 

87 

116 

615 

379 

85 

222 

376 

584 

972 

231 

624 

785 

923 

84 

526 

448 

379 

87 

999 

ADDITION. 


23 


Ex. 

17. 

Add: 

1.  1234 

2.  4321 

3.  2345 

4.  345 

368 

6450 

3456 

2783 

5721 

378 

4567 

1497 

1050 

4291 

5678 

5840 

4862 

5782 

689 

9010 

9215 

6431 

7890 

2709 

5.  5207 

6.  3426 

7.  2358 

8.  9210 

3584 

783 

7291 

1029 

2671 

5279 

5946 

291 

987 

1085 

7368 

3587 

3512 

9270 

5492 

2785 

6705 

876 

876 

8899" 

Ex. 

18. 

Add: 

1.  12345 

2.  23456 

3.     5 

4.  92583 

3275 

72564 

23 

4620 

4721 

3785 

936 

973 

371 

23584 

6543 

25 

51028 

987 

92840 

9 

61234 

96 

72104 

17 

5.  23504 

6.   358 

7.  56789 

8.  123456 

4368 

9246 

3587 

258071 

25 

14376 

296 

589347 

9 

845 

89 

258923 

36 

29 

7 

720145 

378 

7 

12345 

396012 

24 


ADDITION. 


9.  580921 

10.  654321 

11.  '   5 

12.  345 

13.  584321 

42364 

41058 

24 

6197 

92047 

527913 

3792 

358 

52718 

3681 

80235 

589 

1497 

6904 

927 

726048 

75 

36725 

871 

1078 

4386 

9 

187348 

89 

92569 

Ex.  19. 


Add: 

1.  5203461 

2.  2587609 

3.  1357924 

9350472 

3582764 

6804281 

1456849 

1357908 

5975325 

2604030 

4670253 

7101584 

5876543 

8492056 

9276432 

1234567 

4759841 

6789009 

4.  8274108 
3509270 
4680259 
3584672 
9876543 
5279614 


5.  5791350 
246801 
1384650 
2794589 
6532108 
7999888 


38,  It  is  obvious  that  numbers  can  be  added  only  when 
they  refer  to  the  same  things.  Five  oranges  and  three 
books  when  "put  together"  are  still  5  oranges  and  3  books, 
and  not  8  oranges  or  8  books. 

It  is  also  obvious  that  digits  can  be  added  only  when 
they  refer  to  the  same  order  of  units.  Nine  hundreds  and 
eight  tens  when  put  together  are  still  9  hundreds  and  8 
tens,  and  not  17  hundreds  or  17  tens. 


ADDITION. 


25 


Care  must  be  taken,  therefore,  in  writing  numbers  to  be 
added,  that  all  the  units  digits  shall  fall  in  one  column,  all 
the  tens'  digits  in  the  next  column  (to  the  left),  and  all  the 
hundreds  digits  in  the  next  column,  and  so  on. 

39.  To  add  columns  of  digits  with  absolute  accuracy  and 
great  rapidity  is  a  real  accomplishment,  and  the  operation 
of  addition  should  be  continued  until  both  these  results  are 
secured.  The  beginner,  however,  will  need  some  test  of  the 
accuracy  of  his  work.  One  test  is  to  begin  at  the  bottom 
of  the  right-hand  column  in  adding,  and  write  on  a  piece  of 
waste-paper  the  entire  sum  of  each  column ;  then  to  begin 
at  the  top  of  the  left-hand  column  and  write  also  the  entire 
sum  of  each  column ;  finally,  to  add  the  sums  obtained  in 
the  first  addition,  and  the  sums  obtained  in  the  second 
addition,  and  compare  the  results. 

The  study  of  an  example  will  make  the  process  under- 
stood. 


Beginning  at  the  top 

Beginning  at  the  bot- 

of the  left-hand  column 

tom  of  the  right-hand  col- 

in  adding,    and   writing 

871254 

umn  in  adding,  and  writ- 

the  entire  sum  of  each 

123456 

ing  the  entire  sum  of  each 

column,  we  have  : 

789098 

column,  we  have  : 

28 

357912 

26 

31 

993286 

28 

23 

17 

17 

3135006 

23 

28 

31 

26 

28 

3135006 


3135006 


By   comparing    the   results    we   find    each    sum    to    t>e 
3,135,006,  and  so  infer  that  the  operation  is  correct. 


26  ADDITION. 


Find  the  sums  of: 

1.  427,  342,  856,  728. 

2.  483,  1000,  8000,  648,  3750,  9840. 

3.  15,  603,  1145,  6342. 

4  41,  725,  60,  425,  7000,  4900,  398. 

5.  39,  876,  5742,  3000,  478,  9873. 

6.  327,  4960,  5000,  749,  3000,  7849. 

7.  4284,  32,  679,  43,  5006,  7897. 

8.  325,  6007,  983,  4050,  678,  9874. 

9.  856,  9193,  8765,  4287,  6696,  9185,  979. 

10.  7964,  5000,  303,  9784,  5673,  9004. 

11.  9007,  34,  6876,  400,  9344,  7879. 

12.  45,678,  96,  375,  4784,  9673,  11,980. 

13.  7865,  3586,  4321,  8576. 

14.  900,542  +  308,970  +  555,674  +  498,785. 

15.  456,789  +  304,590  +  600,792  +  480,893  +  514,763. 

16.  357,963  +  478,497  +  323,484  +  596,372  +  300,409. 

17.  706,963  +  78,405  +  907,342  +  503,476. 

18.  A  man  bought  a  sleigh  for  $142,  a  carriage  for  $325, 

and  a  pair  of  horses  for  $476.    What  was  the  cost  of 
all? 

19.  A  man  collected  on  Monday,  $  1290 ;  on  Tuesday,  $ 340 ; 

on  Wednesday,  $1008.     How  much  was  collected  in 
all? 

20.  A  lady  paid  $912  for  a  piano,  $342  for  furniture,  $  187 

for  linen,  $46  for  silver.     What  did  she  pay  for  all? 

21.  A  farmer  had  in  one  flock  of  sheep,  407 ;    in  another, 

96 ;  and  in  a  third,  2584.     How  many  had  he  in  all? 


ADDITION.  27 


22.  A  man  owns  four  houses;  the  first  is  worth  $47,050; 

the  second,  $9106;  the  third,  $1492;  the  fourth, 
$  512.  What  is  the  value  of  them  all  ? 

23.  Five  loads  of  flour  weighed  as  follows :  3500  pounds, 

4967  pounds,  3974  pounds,  7982  pounds,  7963 
pounds.  What  was  the  weight  of  the  whole? 

24.  A  house  was  bought  for  $  7895 ;    repairs  amounted  to 

$1500;  new  fences,  $97;  repairs  on  stable,  $463  ; 
furniture,  $1285.  What  was  the  cost  of  the  whole  ? 

25.  The  population  of  six  towns   is :    1674,    9008,  3769, 

4000,  7096,  3784.     Find  the  whole  population. 

26.  A  house-lot  cost  $675 ;  for  building  the  house  and  fur- 

nishing materials  the  carpenters  were  paid  $2245, 
the  masons  $540,  the  painters  $320.  What  was  ex- 
pended on  house  and  lot? 

27.  A  merchant  bought  carpets  to  the  amount  of  $4670 ; 

c.urtains,  $300;  paper-hangings,  $1275;  matting, 
$9765.  What  was  the  cost  of  the  whole  ? 

28.  Find  the  sum  of  three  hundred  thousand  six  hundred 

fifty,  seven  thousand  eight  hundred  thirty-two,  eleven 
thousand  five  hundred  sixty-seven,  ten  thousand  fifty- 
six,  four  hundred  seventy-two. 

29.  Find  the  sum  of   one  hundred  sixty-seven   thousand, 

three  hundred  sixty-seven  thousand,  nine  hundred 
six  thousand,  two  hundred  forty-seven  thousand,  ten 
thousand,  seven  hundred  thousand,  nine  hundred 
seventy-six  thousand,  one  hundred  ninety-five  thou- 
sand, ninety-seven  thousand. 

30.  Find  the  sum    of  two  hundred  seven,  three  hundred 

sixty-two,  nine  hundred  forty-five,  two  thousand 
three  hundred  forty-three,  fifteen  thousand  six  hun- 
dred twenty- two,  forty-five  thousand  eight. 


28  ADDITION. 


31.  Add  3  thousand  4  hundred  92,  one  thousand  four,  6 

thousand  5  hundred  seventy,  42  hundred  eleven. 

32.  Add  386  million  591,  546  million  311  thousand  122, 

796  thousand  351,  84  hundred  1,  9  thousand,  86 
thousand  521,  3  hundred  fifty-eight  thousand  6  hun- 
dred, 8  million  888  thousand  eight  hundred  eighty- 
eight,  1  hundred  million. 

33.  Find  the  sum  of  six  million  sixty  thousand  six,  seven 

million  nine  hundred  fifty  thousand  ninety-nine,  ten 
million  nine  thousand  eight  hundred  seven,  three 
hundred  sixty-seven  thousand  forty-five. 

34.  Find  the  sum  of  200  million  302  thousand,  200  thou- 

sand two  hundred,  50  million  50  thousand  50,  25 
million  860  thousand,  47  million  467  thousand,  202 
million  6367. 

35.  What  is  the  sum  of  eighteen  thousand  three  hundred 

twenty,  seventy-four  thousand  five  hundred  six,  ten 
hundred  seventeen  thousand  nine  hundred  twenty- 
one,  fifty-three  thousand  seven  hundred  eleven,  five 
hundred  seventy-six  thousand  three  hundred  four, 
six  hundred  fifty  thousand  forty-four  ? 

36.  A  man  drew  five  loads  of  bricks ;  in  the  first  load  there 

were  4068 ;  in  the  second,  1342 ;  in  the  third,  3927  ; 
in  the  fourth,  1694 ;  in  the  fifth,  2009.  How  many 
in  all  the  loads? 

37.  What  is  the  united  population  of  the  following  cities : 

Utica,  28,804  ;  Lowell,  40,928 ;  Lynn,  28,236  ;  Sa- 
lem, 24,100;  Erie,  19,500;  Auburn,  17,225? 

38.  A  fruit-grower  sent  to  market  the  produce  of  six  peach 

orchards ;  from  the  first,  7000  baskets ;  from  the 
second,  6973  ;  from  the  third,  1004 ;  from  the  fourth, 
3276;  from  the  fifth,  1594;  from  the  sixth,  3976. 
How  many  baskets  in  all  ? 


ADDITION.  29 


39.  The  distance  from  Boston  to  Springfield  is  98  miles, 

from  Springfield  to  New  Haven  62  miles,  from  New 
Haven  to  New  York  76  miles.  How  many  miles  is 
it  from  Boston  to  New  York  ? 

40.  An  army  officer  paid  at  one  time  $7038  for  horses,  at 

another  time  $7776,  at  another  time  $9948.  How 
many  dollars  did  he  pay  in  all  ? 

41.  A   farmer   sold   his  wheat   for   $8742,  his   corn   for 

$13,569,  and  his  oats  for  $9528.  How  much  did 
he  receive  for  the  whole  ? 

42.  A  bank  has  $40,317  in  specie,  $91,256  in  bills,  $18,317 

in  cash  items.     Find  the  whole  amount. 

43.  The  army  of  Napoleon  at  Waterloo  consisted  of  48,950 

infantry,  15,765  cavalry,  7732  artillery.  What  was 
the  whole  number? 

44.  The  Duke  of  Wellington's  army  at  Waterloo  consisted 

of  20,661  infantry,  8735  cavalry,  6877  artillery. 
There  were  also  33,413  allies.  What  was  the  whole 
number  of  his  army  ? 

45.  The  area  of  England  is  50,535  square  miles,  of  Scot- 

land 29,167  square  miles,  and  of  Wales  8125  square 
miles.  How  many  square  miles  in  England,  Scot- 
land, and  Wales  together  ? 

46.  New   Hampshire    furnished    12,497   soldiers   for  the 

Revolution,  Massachusetts  67,907,  Rhode  Island 
5908,  Connecticut  31,939.  How  many  did  these 
four  states  furnish? 

47.  A  country  merchant  has  in  his  store  flour  worth  $656, 

sugar  worth  $480,  molasses  worth  $325,  cotton 
cloth  worth  $125,  tea  worth  $56,  canned  goods 
worth  $78.  What  is  the  whole  value  of  his  goods? 


30  ADDITION. 


48.  A  farmer  sold  four  loads  of  hay.     The  first  weighed 

2007  pounds,  the  second  1963  pounds,  the  third 
2585  pounds,  the  fourth  2614  pounds.  How  many 
pounds  did  the  whole  weigh  ? 

49.  If  Abraham  was  born  at  the  beginning  of  the  year 

B.C.  1996,  how  many  years  from  the  date  of  his  birth 
to  the  end  of  the  year  1889? 

50.  An  orchard  contains  112  apple  trees,  and  an  equal 

number  of  pear  trees ;  56  peach  trees,  and  an  equal' 
number  of  plum  trees ;  and  19  cherry  trees.  How 
many  trees  are  there  in  the  orchard? 

51.  How  many  times  does  a  clock  strike  from  half  past 

twelve  o'clock  at  night  to  half  past  twelve  o'clock  at 
noon? 

52.  The  area  of  Maine  in  square  miles  is  29,895,  of  New 

Hampshire  9005,  of  Vermont  9135,  of  Massachu- 
setts 8040,  of  Rhode  Island  1085,  of  Connecticut 
4845.  What  is  the  area  of  New  England  in  square 
miles? 

53.  The  area  of  New  York  in  square  miles  is  47,620,  of 

Pennsylvania  44,985,  of  Virginia  40,125,  of  North 
Carolina  48,580,  of  Ohio  40,760.  What  is  the  area 
of  these  five  states  in  square  miles  ? 

54.  The  area  of  Illinois  in  square  miles  is  56,000,  of  Mich- 

igan 57,430,  of  Wisconsin  54,450,  of  Iowa  55,475, 
of  Missouri  68,735.  What  is  the  area  of  these  five 
states  in  square  miles? 

55.  The   area   of  Texas   in   square   miles  is   262,290,  of 

California  155,980,  of  Dakota  147,700,  of  Montana 
145,310,  of  New  Mexico  122,460,  of  Arizona  112,920. 
Find  their  total  area  in  square  miles. 


CHAPTEE  III. 
SUBTRACTION. 

40.  What  number  must  be  added  to  four  to  make  seven? 
What,  then,  will  be  left  if  4  is  taken  from  7  ? 

What  number  must  be  added  to  seven  to  make  ten? 
What,  then,  will  be  left  if  7  is  taken  from  10  ? 

Copy  the  following  set  of  numbers,  and  find  what  num- 
ber must  be  added  to  each  one  in  the  upper  row  to  make 
the  number  below  the  line.  Write  the  required  numbers 
in  the  empty  places  above  the  lines : 

76    12    429056 
17    14    20    7    12    10    5    7    12 

13    19    24    7    13    8    9    4    10 

15    25    28    12    20    10    11    16    21 

When  you  have  done  this,  you  will  see  that,  since  7  and 
10  make  17,  7  taken  from  17  leaves  10;  since  6  and  8 
make  14,  6  taken  from  14  leaves  8 ;  so  with  each  set  of 
numbers. 


32  SUBTRACTION. 


41.  In  the  following  set,  under  each  number  in  the  lower 
row,  write  the  number  that  must  be  added  to  it  to  make 
the  upper  number : 

9        12          7        12        15        10          6          9          7 
342865054 


11        18        17          5        10          9        16          8          3 
52        16          132520 


To  3  we  have  to  add  6  to  make  9,  so  we  write  6  under 
the  3.  To  4  we  must  add  8  to  make  12,  so  we  write  8 
under  the  4. 

Now  in  finding  what  number  must  be  added  to  3  to 
make  9,  we  have  really  found  what  number  will  be  left  if 
3  is  taken  from  9.  In  finding  what  number  must  be  added 
to  4  to  make  12,  we  have  really  found  what  number  will 
remain  if  4  is  taken  from  12. 

42.  The  operation  of  finding  the  number  that  remains, 
when  a  smaller  number  is  taken  from  a  larger,  is  called 
subtraction.     The  result  is  called  the  remainder  or  difference, 

43.  The  number  which  is  to  be  subtracted  is  called  the 
subtrahend;  and  the  number  which  is  to  be  diminished  (that 
is,  the  number  from  which  the  subtraction  is  made),  is  called 
the  minuend. 

44.  A  dash  —  is  the  sign   of    subtraction,    and    when 
placed  between  two  numbers  means  that  the  first  number  is 
to  be  diminished  by  the  second.     It  is  called  the  minus  sign. 

The  expression  4  —  1  =  3  is  read  four  minus  one  equals 
three. 


SUBTRACTION.  33 


45,    Three  dots  .*.  are  often  used  for  the  word  therefore. 
The  expression  6  +  2  =  8,  .'.  8  —  6  =  2,  is  read  six  plus 
two  equals  eight,  therefore  eight  minus  six  equals  two. 

Ex.  21.      (Oral.) 

1.  What  number  with  5  makes  10  ? 
What  number  with  3  makes  10? 
What  number  with  2  makes  10  ? 
What  number  with  4  makes  10? 

2.  What  number  taken  from  10  leaves  2? 
What  number  taken  from  10  leaves  4  ? 
What  number  taken  from  10  leaves  3  ? 
What  number  taken  from  10  leaves  5  ? 

3.  5  is  one  part  of  12,  what  is  the  other? 

8  is  one  part  of  12,  what  is  the  other? 
3  is  one  part  of  12,  what  is  the  other? 

7  is  one  part  of  12,  what  is  the  other  ? 

9  is  one  part  of  12,  what  is  the  other? 
6  is  one  part  of  12,  what  is  the  other? 

10  is  one  part  of  12,  what  is  the  other? 

4.  What  number  taken  from  12  leaves  11  ? 
What  number  taken  from  12  leaves  9  ? 
What  number  taken  from  12  leaves  5  ? 
What  number  taken  from  12  leaves  8  ? 
What  number  taken  from  12  leaves  2  ? 
What  number  taken  from  12  leaves  6  ? 
What  number  taken  from  12  leaves  7  ? 
What  number  taken  from  12  leaves  1  ? 

5.  9  +  2=        .'.11  —  2=        and  11—    9  = 

8  +  3=        .'.11-3=        and  11-    8  = 
6  +  5=        .-.11  —  5=        and  11-    6  = 

10  +  1  =        /.  11  -  1  =        and  11  -  10  = 


34  SUBTRACTION 


6. 

8  +  5  = 

.-.13 

-5  = 

and 

13- 

8 

a 

6  +  7  = 

.\13 

-7  = 

and 

13- 

6 

= 

9  +  4  = 

.-.13 

-4  = 

and 

13- 

9 

= 

7. 

6  +  8^ 

.-.14 

-6  = 

and 

14- 

8 

— 

5  +  9  = 

.\14 

-9  = 

and 

14- 

5 

= 

7  +  7  = 

.-.14 

f-T  

8. 

7  +  8  = 

.-.15 

—  7  = 

and 

15- 

8 

=r 

9  +  6  = 

.-.15 

-6  = 

and 

15- 

9 

tss 

9  +  3  = 

.-.12 

—  9  = 

and 

12- 

3 

sat 

9. 

8  +  8  = 

.-.16 

-8  = 

7  +  9  = 

.-.16 

Y  

and 

16- 

9 

= 

9  +  8  = 

.-.17 

-9  = 

and 

17- 

8 

= 

10.  Subtract  by  threes,  from  100  to  1 ;  from  102  to  0;  by 

fours,  from  101  to  1 ;  from  102  to  2 ;  from  103  to  3. 

11.  Subtract  by  fives,  from  102  to  2 ;  from  103  to  3 ;  from 

104  to  4 ;  from  100  to  5. 

12.  Subtract  by  sixes,  from  103  to  1 ;  from  104  to  2  ;  from 

105  to  3  ;  from  100  to  4 ;  from  102  to  6. 

13.  Subtract  by  sevens,  from  106  to  1 ;  from  100  to  2  ; 

from  101  to  3 ;  from  102  to  4 ;  from  103  to  5  ;  from 

104  to  6 ;  from  105  to  7. 

14.  Subtract  by  eights,  from  105  to  1 ;  from  106  to  2;  from 

107  to  3  ;   from  100  to  4 ;   from  101  to  5  ;   from  102 
to  6 ;  from  103  to  7 ;  from  104  to  8. 

15.  Subtract  by  nines,  from  100  to  1 ;     from  101  to  2 ; 

from  102  to  3 ;  from  103  to  4  ;  from  104  to  5 ;  from 

105  to  6 ;  from  106  to  7. 


SUBTRACTION.  35 


Ex.  22.     (Oral.) 

5  +  4  = 

.-.   9 

-5  = 

9- 

4  = 

9  +  3  = 

.-.12 

9  _ 

12- 

0   

6  +  5  = 

.-.11 

-6  = 

11- 

5  = 

7  +  6  = 

.-.13 

-7  = 

13- 

6  = 

9  +  6  = 

.-.15 

-6  = 

15- 

9  = 

7  +  9  = 

.-.16 

-9  = 

16- 

»7   

14-8=  16-9=  18-6=  17-8=  25-9  = 

11-3=  33-8=  45-6=  76-8=  32-9  = 

16-7=  24-9=  37-8=  48  —  6=  53-9  = 

17-8=  35-8=  43-7=  50-4=  63-6  = 

12-4=  44  —  7=  24-8=  31-3=  26-9  = 

15-7=  68-9=  56-7=  43-5=  29-7  = 

13_6=  27-8=  34-9=  40-9=  50-7  = 

11-8=  13-8=  15-8-  13-9-  31  —  3  = 

27-9=  86-8=  85-9=  87-6=  84-5  = 

32-8=  73-5=  62-7=  26-9=  23-7  = 

25-4=  75-9=  73-7=  72-6=  83-8  = 

17_9=:  31  —  8=  42  —  9=  50  —  3-  39  —  8  = 

42-3=  30-6=  38-9=  40-4-  93-7  = 

37-9=  58-9=  52-6=  63-8=  41-3  = 

24-7=  70-8=  21-9=  22-7=  38-9  = 

45-8=  42-3=  54-7=  71-8=  65-7  = 

19_8  =  60-3=  65-9=  64-6=  17-9  = 

34-6=  95-6=  82-8=  79-9=  76-8  = 

28-9=  72-7=  90-9=  65-6=  81-7  = 

54-5=  77-8=  85-7=  69-9=  71-4- 


36  SUBTRACTION. 


From  876  take  631. 

Write  units  under  units,  tens  under  tens,  and  so  on.     Then  1  unit 

from  6  units  leaves  5  units,  and  we  write  5 

Operation.  under  the  units'  column  ;   3  tens  from  7  tens 

Minuend        876        leave  4  tens,  and  we  write  4  under  the  tens' 

a  U4.    k     A    AQ1         column;  6  hundreds  from  8  hundreds  leave 
bubtranend,  ool 

2.  hundreds,  and  we  write  2  under  the  nun- 
Remainder,    245        dreds'  column.     The  remainder,  therefore,  is 
2  hundreds  4  tens  5  units ;  that  is,  245. 

46,  The  minuend  is  the  sum  of  the  subtrahend  and  the 
remainder.     Hence,  to  test  the  accuracy  of  the  work,  add 
the  subtrahend  and  remainder  together,  and  if  the  work  is 
correct,  their  sum  will  be  equal  to  the  minuend. 

47,  It  is  obvious  that  one  number  can  be  subtracted  from 
another  only  when  both  numbers  refer  to  the  same  things. 
Thus,  we  can  subtract  3  oranges  from  5  oranges,  but  we 
cannot  subtract  3  apples  from  5  oranges. 

Ex.  23. 
Find  the  results  of: 


1. 

59- 

23. 

13. 

89- 

41. 

25. 

786- 

45. 

2. 

54- 

23. 

14. 

67- 

23. 

26. 

674- 

52. 

3. 

67- 

14. 

15. 

58- 

17. 

27. 

569- 

38. 

4. 

65- 

32. 

16. 

75- 

34. 

28. 

857- 

43. 

5. 

78- 

25. 

17. 

96- 

53. 

29. 

294- 

82. 

6. 

75- 

41. 

18. 

87- 

42. 

30. 

348- 

37. 

7. 

85- 

33. 

19. 

69- 

37. 

31. 

489- 

76. 

8. 

78- 

25. 

20. 

78- 

26. 

32. 

768- 

47. 

9. 

96- 

42. 

21. 

64- 

43. 

33. 

976- 

53. 

10, 

97- 

54. 

22. 

98- 

35. 

34. 

897- 

75. 

11. 

87- 

54. 

23. 

89- 

53. 

35. 

588- 

64. 

12. 

86  — 

31. 

24. 

77- 

46. 

36. 

467- 

45. 

SUBTRACTION. 


37 


37.  874  -  632.  42.  6982  -  5431.  47.  725,419  -  613,208. 

38.  792  -  261.  43.  7629  -  4518.  48.  965,420  -  342,100. 

39.  798  -  627.  44.  7824  -  6821.  49.  854,267  -  723,150. 

40.  764  -  532.  45.  8542  -  6131.  50.  549,830  -  438,820. 

41.  862  -  741.  46.  8792  -  6281.  51.  628,300  -  517,200. 


48,  If  the  number  of  units  of  any  order  in  the  minuend 
is  less  than  the  number  of  units  of  the  corresponding  order 
in  the  subtrahend,  one  of  the  next  higher  order  of  units  in 
the  minuend  must  be  added  to  the  units  of  the  order  we 
are  considering.  The  process  will  be  understood  by  an 
example. 

From  783  take  469. 


Since  we  cannot  take  9  units  from  3  units,  we  add  1  of  the  8  tens 
to  the  3  units,  making  13  units;  then  9  units 
from  13  units  leave  4  units.  Now  as  we  have 
added  1  of  the  8  tens  to  the  3  units  of  the  min- 
uend, we  have  only  7  tens  remaining,  and  6 
tens  from  7  tens  leave  1  ten ;  4  hundreds  from 
7  hundreds  leave  3  hundreds.  The  remainder, 
therefore,  is  3  hundreds  1  ten  4  units ;  that  is,  314. 


Operation. 
Minuend,  783 
Subtrahend,  469 
Remainder,  314 


From  359  take  186. 

Here  6  units  from  9  units  leave  3  units.  Since  we  cannot  take  8 
.  tens  from  5  tens  we  add  1  of  the  3  hundreds  to 

on'  the  5  tens,  making  15  tens;  then  8  tens  from 

Minuend,       359    15  tens  leave  7  tens>     Now  as  we  have  added 

1  of  the  3  hundreds  to  the  5  tens  of  the  minu- 
end, we  have  only  2  hundreds  remaining ;  and 
1  hundred  from  2  hundreds  leaves  1  hundred. 


Subtrahend,  186 
Remainder,    173 


The  remainder,  therefore,  is  1  hundred  7  tens  3  units  ;  that  is,  173. 


38  SUBTRACTION. 


Ex.  24. 

1. 

867 

-325. 

13. 

90- 

35. 

25. 

70  -  28. 

2. 

985 

-312. 

14. 

40- 

13. 

26. 

50  -  13. 

3. 

746 

-213. 

15. 

70- 

26. 

27. 

80  -  37. 

4. 

384 

-  132. 

16. 

50- 

24. 

28. 

60  -  48. 

5. 

479 

-235. 

17. 

80- 

32. 

29. 

90- 

-25. 

6. 

679 

-215. 

18. 

60- 

33. 

30. 

50  -  27. 

7. 

857 

-324. 

19. 

60- 

47. 

31. 

80  -  43. 

8. 

956 

-532. 

20. 

70- 

45. 

32. 

70-36. 

9. 

795 

-362. 

21. 

70- 

52. 

33. 

90  -  32. 

10. 

687 

-321. 

22. 

80- 

36. 

34. 

60  -  27. 

11. 

978 

-333. 

23. 

90- 

28. 

35. 

80-49. 

12 

835 

-214. 

24. 

90- 

27. 

36. 

90  -  36. 

Ex.  25. 

1. 

5.2-26. 

13. 

63- 

29. 

25. 

680 

-247. 

2. 

73  -  38. 

14. 

74- 

37. 

26. 

570 

-236. 

3. 

81  -  49. 

15. 

92- 

68. 

27. 

860 

-218. 

4. 

94-57. 

16. 

81- 

56. 

28. 

690 

-254. 

5. 

72  -  48. 

17. 

75- 

38. 

29. 

750 

-419. 

6. 

91-64. 

18. 

96- 

48. 

30. 

830 

-214. 

7. 

75- 

-48. 

19. 

85- 

57. 

31. 

690 

-275. 

8. 

92  -  48. 

20. 

93- 

75. 

32. 

750 

-326. 

9. 

83- 

-26. 

21. 

^4 
«j  i 

18. 

33. 

860 

-247. 

10. 

95  -  47. 

22. 

81- 

27. 

34. 

970 

-358. 

11. 

86  -  57. 

23. 

75- 

29. 

35. 

580 

-149. 

12. 

95  —  66. 

24. 

94- 

58. 

36. 

870 

-146. 

Ex.  26. 

1. 

407 

-84. 

7. 

462-38. 

13. 

608 

—  247. 

2. 

308 

-75. 

8. 

374-57. 

14. 

706 

-253. 

3. 

609 

-58. 

9. 

281  —  65. 

15. 

805 

-364. 

4. 

205 

-81. 

10. 

592  -  83. 

16. 

904 

-472. 

5. 

506 

-63. 

11. 

476  -  68. 

17. 

809 

-581. 

6. 

807 

-42. 

12. 

852  -  39. 

18. 

705 

-r694. 

SUBTRACTION.  39 


19. 

508 

-294. 

25. 

781 

-  246. 

31. 

461- 

239. 

20. 

609 

-385. 

26. 

892 

-387. 

32. 

572  — 

238. 

21. 

707 

-246. 

27. 

643 

-418. 

83. 

693- 

447, 

22. 

806 

-324. 

28. 

954 

-  216. 

34. 

754- 

536. 

23. 

405 

-132. 

29. 

763 

-419. 

35. 

835- 

226. 

24. 

709 

-328. 

30. 

655 

-247. 

36. 

973- 

237. 

Ex. 

27. 

1. 

612 

-78. 

13. 

732 

-458. 

25. 

531  — 

352. 

2. 

523 

-64. 

14. 

816 

-237. 

26. 

642- 

263. 

3. 

845 

-87. 

15. 

624 

-158. 

27. 

763- 

174. 

4. 

417 

-58. 

16. 

936 

-489. 

28. 

824- 

296. 

5. 

731 

-94. 

17. 

567 

-298. 

29. 

915- 

468. 

6. 

324 

-65. 

18. 

715 

-348. 

30. 

812- 

357. 

7. 

942 

-74. 

19. 

623 

-417. 

31. 

514- 

136. 

8. 

635 

-89. 

20. 

861 

-375. 

32. 

972- 

489. 

9. 

522 

-56. 

21. 

453 

-  286. 

33. 

624- 

248. 

10. 

417 

-68. 

22. 

817 

-329. 

34. 

512- 

136 

11. 

325 

-86. 

23. 

643 

-457. 

35. 

713- 

364. 

12. 

712 

-94. 

24. 

415 

-186. 

36. 

817- 

259 

Ex. 

28. 

1. 

500 

-78. 

13. 

600 

—  235. 

25. 

902  — 

146. 

2. 

600 

-83. 

14. 

800 

-217. 

26. 

805- 

347. 

3. 

700 

-92. 

15. 

900 

-386. 

27. 

704- 

215. 

4. 

800 

fi4 

\J  Jt. 

16. 

700 

-  427. 

28. 

607- 

238. 

5. 

600 

-57. 

17. 

400 

-128. 

29. 

503  — 

267. 

6. 

400 

-76. 

18. 

800 

-372. 

30. 

906- 

387. 

7. 

802 

-68. 

19. 

600 

-345. 

31. 

904- 

328. 

8. 

304 

-95. 

20. 

700 

-562. 

32. 

802- 

467. 

9. 

506 

-87. 

21. 

800 

-427. 

33. 

705- 

258. 

10. 

403 

-75. 

22. 

900 

-368. 

34. 

603- 

318. 

11. 

902-94. 

23. 

500 

-321. 

35. 

701- 

427. 

12. 

504 

-69. 

24. 

600 

-487. 

36. 

705- 

348. 

40  SUBTRACTION. 


Ex.  29. 

1.  7689-2345.  9.  9580  —  5136.  17.  8300-2746. 

2.  6837-4216.  10.  7480-2367.  18.  7400-2843. 

3.  9876  -  1234.  11.  9560  -  1423.  19.  8020  -  3647. 

4.  8697-3274.  12.  8670-4324.  20.  7050-6873. 

5.  7586-2145.  13.  8700-3218.  21.  6040-2895. 

6.  6789-4321.  14.  9600-2745.  22.  8030-2746. 

7.  8470-2138.  15.  9600-4347.  23.  7050-4873. 

8.  6790-3245.  16.  7200-3647.  24.  6020-2748. 

Ex.  30. 

1.  6005-2347.  9.  8021-3472.  17.  9000-3725. 

2.  8002-2636.  10.  8064-2397.  18.  9000-2745. 

3.  8003-2746.  11.  9012-3684.  19.  6324-2538. 

4.  6005  —  2748.  12.  7054-2768.  20.  6245-3789. 

5.  9004-2615.  13.  7000-2546.  21.  4517-1638. 

6.  6003-2846.  14.  7000-3748.  22.  7253-4867. 
7.7035-2648.  15.  8000-5318.  23.  9215-4757. 
8.  7023-2896.  16.  8000-3526.  24.  7214-4869. 

Ex.  31. 

1.  56,739-24,316.  13.  59,001-16,739. 

2.  68,507-47,623.  14.  89,076-   569. 

3.  47,865-12,341.  15.  60,020-24,156. 

4.  72,006-48,315.  16.  57,490-   598. 

5.  65,043-17,872.  17.  70,000-25,487. 

6.  81,000-25,143.  18.  70,000-  4,139. 

7.  90,000-30,906.  19.  60,300-36,428. 

8.  90,503-47,628.  20.  70,302-  5,648. 

9.  41,009-31,214.  21.  80,040-23,619. 

10.  43,020-36,748.       22.  63,008-47,236. 

11.  26,735-  9,856.       23.  50,004-47,825. 

12.  75,986-43,264.      24.  80,047-26,578. 


SUBTRACTION.  41 


Ex.  32. 

1.  431,250  —  153,697.  8.  842,003-459,687. 

2.  920,503-476,829.  9.  715,324-369,857. 

3.  523,146-286,759.  10.  900,500-465,783. 

4.  647,352-268,574.  11.  512,435-126,867. 

5.  502,304—186,475.  12.  600,000-285,436. 

6.  625,030-274,384.  13.  723,514-536,945. 

7.  720,301-368,596.  14.  801,050-469,872. 

Ex.  33. 

1.  What  number  must  be  added  to  7428  to  make  8047? 

2.  What  number  must  be  taken  from  3015  to  leave  2405? 

3.  If  the  minuend  is  78,206,  and  the  subtrahend  35,264, 

what  is  the  remainder  ? 

4.  A  man  owed  $4689.     He  paid  at   one   time   $3894. 

How  much  did  he  still  owe  ? 

5.  A  flour  merchant  had  on  hand  2038  barrels  of  flour, 

He  sold   1299   barrels.      How   many   barrels   had 
he  left? 

6.  Mr.  Brown's  yearly  income  is  $5067.      His  expenses 

are  $4093.     How  much  does  he  save? 

7.  The  population  of  New  England  in  1870  was  3,487,924, 

in  1880,  4,010,529.     Find  the  increase. 

8.  A  house  cost  $9468.     If  payments  to  the  amount  of 

$5889  have  been  made  to  the  builder,  how  much 
still  remains  due? 

9.  The  sum  of  two  numbers  is  890,375,  and  one  of  them  is 

309,007.     What  is  the  other  ? 


42  SUBTRACTION. 


10.  A  is  worth  $98,760 ;  B  is  worth  $4586  less  than  A. 

How  much  is  B  worth? 

11.  In  1880  the  population  of  Boston  was  369,832,  and 

the  population  of  Baltimore  was  332,313.  How 
much  greater  was  the  population  of  Boston  than 
that  of  Baltimore  ? 

12.  A  tank  holding  370  gallons  of  water  was  filled  by 

pouring  77  gallons  into  it.  How  many  gallons 
were  there  already  in  the  tank  ? 

13.  What  number  increased  by  15,639  will  be  28,984  ? 

14.  What  number  subtracted  from  nine  hundred  eighty- 

seven  thousand  three  hundred  fifty-nine  will  leave 
three  hundred  thousand  two  hundred  eight? 

15.  A  cotton  planter  raised  9675  pounds  of  cotton.     He 

sold  7876  pounds.     How  many  pounds  had  he  left  ? 

16.  There  were  322  apples  on  a  tree,  of  which  198  were 

gathered,  and  87  were  blown  off  by  the  wind.  How 
many  were  left  on  the  tree  ? 

17.  There  are  60  minutes  in  an  hour;  how  many  minutes 

between  4  minutes  after  10  o'clock  and  3  minutes 
before  11  o'clock  ?  Between  9  minutes  after  1 
o'clock  and  3  minutes  before  2  o'clock? 

18.  A  man  purchases  a  farm  for  $24,669,  and  pays  down 

$13,708.     How  much  remains  unpaid? 

19.  Eight   hundred   seventy-six   thousand   four  hundred 

twenty-five  added  to  a  certain  number  makes 
eleven  million  seven  hundred  nine  thousand  three 
hundred  four.  What  is  the  number? 


SUBTRACTION.  43 


20.  Two  men,  A  and  B,  start  together  from  the  same  place 

and  travel  in  the  same  direction.  A  walks  the 
first  day  29  miles,  B  rides  the  first  day  67  miles. 
How  many  miles  apart  are  they  at  the  end  of  the 
first  day  ?  How  many  miles  would  they  have  been 
apart  if  they  had  travelled  in  opposite  directions  ? 

21.  A  merchant  deposited  in  a  bank  $10,040;  and  after- 

wards drew  a  check  for  $3780.  How  much  had  he 
in  the  bank  after  the  check  was  paid  ? 

22.  What  is  the  difference  between  106,074  and  28,999  ? 

23.  A  man  went  to  market  with  $10.25.     He  paid  for 

steak  $2;  for  sugar,  $1  ;  for  coffee,  $1;  for  fruit, 
$2  ;  for  flour,  $2.  How  much  money  had  he  left? 

24.  A  horse  cost  $397  and  was  sold  for  $563.    How  much 

was  gained? 

25.  A  horse  and  carriage  were  bought  for  $458,  and  were 

sold  for  $539.     What  was  the  gain  ? 

26.  A  cow  was  sold  for  $171.25.     The  cow  cost  $152. 

What  was  the  gain  ? 

27.  A  man  bought  a  house  lot  for  $1290  and  sold  it  for 

$  1196.     How  much  did  he  lose  ? 

28.  A  horse,  harness,  and  saddle  were  bought  for  $378, 

and  were  sold  for  $423.50.    How  much  was  gained? 

29.  A  man  owing  $7862.50  has  paid  $5678.     How  much 

is  still  due  ? 

30.  From  a  $50  bank-note   a  bill  of  $38.50  was  paid. 

What  change  was  given  back  ? 

31.  In  the  siege  of  Gibraltar  (1779-1783)  the   English 

fired  57,163  round  shot,  and  the  French,  175,741. 
How  many  more  did  the  French  fire  than  the 
English  ? 


44  SUBTRACTION. 


32.  The  length  of  the  Missouri  River  from  its  source  to  the 

Mississippi  is  three  thoasand  ninety-six  miles,  and 
from  its  source  to  the  Gulf  of  Mexico  four  thousand 
five  hundred  six  miles.  How  many  miles  is  it  from 
the  junction  of  the  two  rivers  to  the  Gulf  of  Mexico  ? 

33.  A  lady  bought  articles  in  a  store  amounting  to  nine 

dollars  and  seventy-five  cents.  She  gave  in  pay- 
ment a  ten-dollar  bill.  How  much  change  should 
she  receive? 

34.  A  gentleman  received  from  his  father  $65,784.     He 

paid  for  a  house  $28,598.     How  much  had  he  left? 

35.  If  the  area  of  the   Mississippi   Valley   is   1,237,111 

square  miles,  and  the  area  of  the  Atlantic  slope  is 
967,576  square  miles,  find  the  excess  of  the  Missis- 
sippi Valley  over  the  Atlantic  slope  in  square  miles. 

36.  Lake  Erie  covers  9600  square  miles,  and  Massachusetts 

contains  8040  square  miles.  How  many  more  square 
miles  in  Lake  Erie  than  in  Massachusetts? 

37.  The  foreign  immigration  into  the  United  States  was, 

in  1883,  603,322,  and  in  1885,  395,346.  How 
much  greater  was  the  number  in  1883  than  in  1885  ? 

38.  The   consumption   of  imported   sugar  in  the  United 

States  was,  in  1882,  866,517  tons,  and  in  1880, 
730,519  tons.  How  many  more  tons  were  con- 
sumed in  1882  than  in  1880? 

39.  A  lady  bought  goods  amounting  to  two  dollars  and 

thirty-four  cents.  She  gave  a  five-dollar  bill  in  pay- 
ment, What  change  should  she  receive  ? 

40.  The  polar  diameter  of  the  earth  is  41,707,620  feet,  and 

the  equatorial  diameter  is  41,847,426  feet.  Find 
the  difference  in  feet. 


SUBTRACTION.  45 


41,  The  Secretary  of  the  Treasury  of  the  United  States 
estimated  the  revenue  for  1885  to  be  $330,000,000. 
The  actual  revenue  was  $323,690,706.  How  much 
did  the  actual  fall  short  of  the  estimated  revenue  ? 

42  The  population  of  Chicago  in  1860  was  109,260,  in 
1880,  503,185.  Find  the  increase. 

43.  The  gross  earnings  of  the  Eastern  Railroad  for  1883 

were  $3,584,506,  and  the  expenses  were  $2,310,830. 
Find  the  net  earnings  for  the  year. 

44.  The   population   of    New   York    City   in    1880   was 

1,206,299,  in  1860  it  was  805,651.  Find  the  in- 
crease for  twenty  years 

45.  In  1880  Kentucky  raised  149,017,855  pounds  of  tobacco, 

and  Virginia  raised  78,421,860  pounds.  How  many 
more  pounds  did  Kentucky  raise  than  Virginia  ? 

46.  The  value  of  the  tobacco  raised  in  Kentucky  in  1880 

was  $10,431,250,  the  value  of  that  raised  in  Virginia 
was  $6,273,749.  Find  the  difference. 

47.  The  population  of  Massachusetts  in  1880  was  1,783,085, 

and  of  Virginia  1 .512  565.     Find  the  difference. 

48.  The  population  of  New  York  in  1880  was  5,082,871, 

and  of  Ohio  3,198,062.     Find  the  difference. 

49.  The   population  of  the   United   States   in    1880  was 

50,155,783,  in  1870,  38,558,371.     Find  the  increase. 

50.  In   1885   the  railroads   of  the   United  States  earned 

from  freight  $519,690,992,  and  from  passengers 
$200.883,911.  How  much  more  was  earned  from 
freight  than  from  passengers? 

51.  The  imports  of  raw  cotton  into  England  in  1871  were 

1,778,139,776  pounds,  the  exports  were  362,075,616 
pounds.  How  many  more  pounds  were  imported 
than  exported  ? 


CHAPTER  IV. 

MULTIPLICATION. 

49,  If  the  cost  of  G  tons  of  coal  at  $  7  a  ton  is  required, 
the  amount  can  be  found  by  writing  $7  six  times  in 

a  column  convenient  for  adding,  as  in  the  margin,         ^L 
and  finding  the  sum  of  the  column.  7 

7 

50,  If  the  cost  of  a  whole  cargo  of  coal  was  re-  7 

quired,  this  operation  would  be  long  and  tedious,  7 

and  therefore  a  shorter  process  has  been  devised, 
called  Multiplication, 

By  this  process  $7  is  written  only  once,  6  is  written  be- 
neath the  $7  to  show  the  number  of  times  $7  must  be 
taken  in  order  to  obtain  the  required  amount,  and  this 
amount  is  found  by  saying  6  times  $  7  are  $42.  Thus : 

o>7 
$< 

6 
$42 

51,  In  this  operation  $7  is  called  the  multiplicand,  6  the 
multiplier,  and  $42  the  product,     The  multiplier  6  is  the 
sum  of  six  1's,  and  the  product  42  is  the  sum  of  six  7's. 
Hence,  it  will  be  seen  that : 

Multiplication  is  an  operation  by  which,  when  two  num- 
bers are  given,  called  multiplicand  and  multiplier,  a  third 
number  is  found  called  product,  which  is  formed  from  the 
multiplicand  as  the  multiplier  is  formed  from  unity. 


MULTIPLICATION.  47 


52,  The  multiplicand  is  the  number  to  be  multiplied. 
The  multiplier  is  the  number  by  which  we  multiply.     The 
product  is  the  result  obtained.     The  multiplicand  and  mul- 
tiplier are  called  factors  of  the  product.     The 
product  of  two  or  more  factors  is  the  same  in 
whatever  order  they  are  taken.     Thus,  3x4 
—  4  X  3.     The  dots  in  the  margin,  read  hori- 
zontally, make  3  fours ;  read  vertically,  make  4  threes. 

53,  The  sign  of  multiplication  is  X.     When  the  multi- 
plier precedes  the  multiplicand,  the  sign  X  is  read  times. 
Thus,  6  X  $7  —$42  is  read  6  times  $7  equal  $42. 

54,  When  the  multiplier  follows  the  multiplicand  the 
sign  X  is  read  multiplied  ly.     Thus,  $7  X  6  =$42  is  read 
$7  multiplied  by  6  equal  $42;  and  means  $7   taken   6 
times  equal  $42.     In  all  cases  the  product  refers  to  the 
same  kind  of  units  as  the  multiplicand. 

55,  Products  of  two  factors,  which  are  each  less  than 
ten,  must  be  learned  by  heart. 

They  can  all  be  readily  found  by  addition.  Thus,  if  the 
product  of  4  times  6  is  required,  we  see  thaj  the  multiplier 
4  is  the  sum  of  four  1's,  and  the  multiplicand  is  6;  hence, 
the  product  is  the  sum  of  four  6's,  and  we  write 

6 
6 

6      Thus,  4  X  6  =  24. 
J> 

24 

In  the  same  way  every  product  is  found  when  each  of 
its  two  factors  is  less  than  ten;  and  the  results  are  all 
written  in  the  following  multiplication  table  : 


48 


MULTIPLICATION. 


MULTIPLICATION   TABLE. 


2 

3 

4 

5 

TIMES 

TIMES 

TIMES 

TIMES 

1  ARE     2 

1  ARE     3 

1  ARE     4 

1  ARE     5 

2  ARE     4 

2  ARE     6 

2  ARE     8 

2  ARE  10 

3  ARE     6 

3  ARE     9 

3  ARE  12 

3  ARE  15 

4  ARE     8 

4  ARE  12 

4  ARE  16 

4  ARE  20 

5  ARE  10 

5  ARE  15 

5  ARE  20 

5  ARE  25 

6  ARE  12 

6  ARE  18 

6  ARE  24 

6  ARE  30 

7  ARE  14 

7  ARE  21 

7  ARE  28 

7  ARE  35 

8  ARE  16 

8  ARE  24 

8  ARE  32 

8  ARE  40 

9  ARE  18 

9  ARE  27 

9  ARE  36 

9  ARE  45 

6 

7 

8 

9 

TIMES 

TIMES 

TIMES 

TIMES 

1  ARE     6 

1  ARE      7 

1  ARE     8 

1  ARE     9 

2  ARE  12 

2  ARE  14 

2  ARE  16 

2  ARE  18 

3  ARE  18 

3  ARE  21 

3  ARE  24 

3  ARE  27 

4  ARE  24 

4  ARE  28 

4  ARE  32 

4  ARE  36 

5  ARE  30 

5  ARE  35 

5  ARE  40 

5  ARE  45 

6  ARE  36 

6  ARE  42 

6  ARE  48 

6  ARE  54 

7  ARE  42 

7  ARE  49 

7  ARE  56 

7  ARE  63 

8  ARE  48 

8  ARE  56 

8  ARE  64 

8  ARE  72 

9  ARE  54 

9  ARE  63 

9  ARE  72 

9  ARE  81 

MULTIPLICATION.  49 


Ex.  34.     (Oral.) 

1.  Of  what  number  are  2  and  4  the  factors?  3  and  3? 

5  and  3?  2  and  5?  3  and  6? 

2.  What  are  the  factors  of  14?  of  9?  of  8?  of  18?  of  6? 

of  21?  of  10? 

3.  4  is  one  factor  of    8  ;  what  is  the  other  ? 

3  is  one  factor  of  12 ;  what  is  the  other  ? 
9  is  one  factor  of  18 ;  what  is  the  other  ? 

4.  3  times  what  number  make  15? 

6  times  what  number  make  18? 

5  times  what  number  make  25  ? 

4  times  what  number  make  28  ? 
3  times  what  number  make  21  ? 

7  times  what  number  make  14  ? 

5.  8  times  what  number  make  24  ? 

6  times  what  number  make  24  ? 

6  times  what  number  make  12?  42?  30?  18? 

6.  7  times  what  number  make  21?  63?  35?  49?  56?  14? 

7.  8  times  what  number  make  32?  64?  16?  40?  24?  56? 

8.  9  times  what  number  make  27?  72?  45?  63?  36?  18? 

9.  6x2—      10.    7x3^-      11.    8x4=      12.    0  X  5  = 


8x2  = 

9x3  = 

1x4  = 

3x5  = 

3x2  = 

0x3  = 

0x4  = 

2x5- 

9x2  = 

4x3  = 

3x4  = 

7x5- 

7x2  = 

3x8  = 

9x4  = 

1x5  = 

1x2  = 

3x3  = 

7x4  = 

8x5  = 

0x2  = 

3x0  = 

4x4  = 

6x5  = 

5x2  = 

3x5  = 

6x4  = 

4x5  = 

4x2  = 

3x1  = 

4x4- 

9x5  = 

50  MULTIPLICATION. 


13.  1x6  = 

14.  3x7  = 

15.  4x8  = 

16.  1x9  = 

9x6  = 

0x7  = 

0x8  = 

7x9  = 

8x6  = 

1x7  = 

3x8  = 

0x9  = 

3x6  = 

2x7  = 

9x8  = 

3x9  = 

5x6  = 

4x7  = 

7x8  = 

8x9  = 

0x6  = 

8x7  = 

1x8  = 

6x9  = 

7x6  = 

5x7  = 

5x8  = 

9X9  = 

2x6  = 

9x7  = 

8x8  = 

5x9  = 

4x6  = 

6x7  = 

8x6  = 

9x9  = 

17.  4x6- 

18.  3x2  = 

19.  5x2  = 

20.  7x4  = 

7x3  = 

7x9  = 

8x2  = 

8x8  = 

9x2  = 

8x3  = 

6x4  = 

0x2  = 

5x3  = 

4x5  = 

7x3  = 

1x9  = 

7x4  = 

9x6  = 

0x9  = 

6x5  = 

8x2  = 

7x4  = 

7x6  = 

7x7  = 

5x7  = 

8x9  = 

8x5  = 

9x9  = 

9x3  = 

6x5  = 

9x5  = 

4x8  = 

4x9  = 

7x8  = 

3x6  = 

7x2  = 

56,  When  the  multiplicand  consists  of  two  or  more 
digits,  and  the  multiplier  is  a  single  digit,  it  is  necessary  to 
multiply  each  digit  of  the  multiplicand  by  the  multiplier. 
Thus,  the  product  of  6  X  4587  is  the  sum  of  six  numbers, 
each  the  same  as  the  multiplicand. 

The  sum  of  the  six  7's  is  6  times  7  =  42,  and  we  write  the  2  units 
in  the  column  of  units,  and  reserve  the  4  tens  to 
be  added  to  the  product  of  the  tens  ;  then  6  times     4587 
8  tens  =  48  tens,  which,  with  the  4  tens,  make     4587 
52  tens,  or  5  hundreds  and  2  tens,  and  we  write     4587  . 

the  2  tens  in  the  column  of  tens ;  then  6  times  5     4587        

hundreds  =  30  hundreds,  which,  with  the  5  hun-     4587        27522 
dreds,  make  35  hundreds,  or  3  thousands  and  5     4587 
hundreds,  and  we  write  the  5   hundreds  in  the 
column  of  hundreds  :  then  6  times  4  thousands  =  24  thousands,  which, 
with  the  3  thousands,  make  27  thousands,  and  we  write  27  to  the  left 
of  the  5  hundreds. 


MULTIPLICATION.  51 


57,  When  the  multiplier  is  10,  100,  1000,  etc.,  the  pro- 
duct is  obtained  by  simply  annexing  as  many  zeros  to  the 
multiplicand  as  are  found  in  the  multiplier.  Thus  : 

10  x  4587  =  45,870. 

Likewise,  when  the  multiplier  is  any  one  cf  the  nine 
significant*  digits  followed  by  zeros,  the  product  is  obtained 
by  multiplying  the  multiplicand  by  the  significant  digit 
and  annexing  to  the  result  as  many  zeros  as  are  found  in 
the  multiplier.  Thus,  if  the  multiplicand  is  4587,  and  the 
multiplier  is  600,  we  multiply  by  6  and  obtain  27,522,  and 
annex  to  this  result  2  zeros,  and  have  for  the  required  pro- 
duct 2,752,200 :  458? 

600 
2,752,200 

Ex.  35. 
Find  the  products  of: 


1. 

4x 

80. 

13. 

6x 

32. 

25. 

3x 

97. 

37. 

9x 

96. 

2. 

8X 

40. 

14. 

2x 

62. 

26. 

8x 

57. 

38. 

6x 

59. 

3. 

9x 

70. 

15. 

7  x 

47. 

27. 

8x 

75. 

39. 

4x 

83. 

4. 

7x 

60. 

16. 

3x 

53. 

28. 

9x 

74. 

40. 

7x 

84. 

5. 

5x 

60. 

17. 

8x 

54. 

29. 

9x 

28. 

41. 

5x 

94. 

6. 

9x 

80. 

18. 

4x 

87. 

30. 

2x 

86. 

42. 

8x 

96. 

7. 

6x 

90. 

19. 

9x 

63. 

31. 

2x 

67. 

43. 

8x 

86. 

8. 

9x 

40. 

20. 

5x 

96. 

32. 

3x 

95. 

44. 

9x 

78. 

9. 

7  x 

40. 

21. 

5x 

78. 

33. 

7x 

85. 

45. 

7x 

53. 

10. 

5x 

90. 

22. 

6x 

58. 

34. 

4x 

79. 

46. 

8x 

83. 

11. 

8x 

50. 

23. 

4x 

86. 

35. 

Sx 

74. 

47. 

9x 

68. 

12. 

5x 

70. 

24. 

7x 

89. 

36. 

5x 

68. 

48. 

7x 

94. 

•  The  digits  1,  2,  3,  4,  5,  6,  7,  8,  9  are  called  significant  digits. 


52  MULTIPLICATION. 


1. 

2. 

Find  the 
7  X  800. 
4  x  200. 

Ex. 
products  off 
13.  6  X  703. 
14.  9  x  507. 

36. 

25.  5  x  974. 
26.  4  x  789. 

37 

38 

8  x  948. 
9  x  827. 

3. 

9 

X 

700. 

15.  5 

X809. 

27.  4  x  947. 

39 

7  x  825. 

4. 

5 

X 

300. 

16.  7 

X604. 

28.  5x987. 

40 

. 

8  x  493. 

5. 

8 

X 

600. 

17.  4 

X906. 

29.  6  x  896. 

41 

9  x  672. 

6. 

7 

X 

400. 

18.  6 

X803. 

30.  6x456. 

42 

. 

7  x  756. 

7. 

6 

X 

750. 

19.  2 

X986. 

31.  7x627. 

43 

. 

8  x  359. 

8. 

4 

X 

340. 

20.  2 

X593. 

32.  7  X  645. 

44 

6  x  387. 

9. 

M 

I 

X 

960. 

21.  3 

X593. 

33.  5x865. 

45 

9  X  865. 

10. 

6 

X 

580. 

22.  3 

X486. 

34.  8x329. 

46 

5  x  739. 

11. 

8 

X 

680. 

23.  4 

X867. 

35.  6x496. 

47 

9  x  648. 

12. 

8 

X 

630. 

24.  3 

X837. 

36.  9  x  584. 

48 

• 

4  X  867. 

Ex. 

37. 

Find  the 

products  of: 

1. 

9 

X 

6000. 

13.  8 

X 

6070. 

25. 

2  X  6007. 

2. 

4 

X 

8000. 

14.  4 

x 

9080. 

26. 

9 

X  7008. 

3. 

7 

X 

8000. 

15.  6 

X 

5080. 

27. 

3 

X  8005. 

4. 

7 

X 

9000. 

16.  7 

X 

4070. 

28. 

8 

X  4007. 

5. 

8 

X 

6000. 

17.  3 

X 

7040. 

29. 

4 

X  6009. 

6. 

6 

X 

7000. 

18.  9 

X 

3050. 

30. 

7 

X  5006. 

7. 

6 

X 

7300. 

19.  9 

X 

6320. 

31. 

7 

X  8026. 

8. 

6 

X 

7400. 

20.  7 

X 

3980. 

32. 

6 

X  7054. 

9. 

7 

X 

8500. 

21.  6 

X 

8570. 

33. 

5 

X  9045. 

10. 

6 

X 

8600. 

22.  5 

X 

7390. 

34. 

4 

X  6072. 

11. 

5 

X 

3900. 

23.  6 

X 

8570. 

35. 

0 

X  6038. 

12. 

7 

X 

7500. 

£4.  8 

X 

6780. 

36. 

5 

X  5076. 

MULTIPLICATION.  53 

Ex.  38. 

Find  the  products  of: 

1.7x7204.               13.2x4716.  25.6x3725. 

2.  3x6305.               14.  3x3825.  26.  7x5273. 

3.  8  X  9308.      15.  4  x  6918.  27.  8  x  6531. 

4.  6  x  4706.      16.  5  x  5724.  28.  9  x  1365. 

5.  4  x  6407.      17.  6  x  6375.  29.  2  x  8417. 

6.  9  x  3809.      18.  7  x  8413.  30.  3  x  7148. 

7.  7  X  3628.      19.  8  X  5823.  31.  4  x  6528. 

8.  8  x  6984.      20.  9  X  3285.  32.  5  X  8256. 

9.  8  X  5746.      21.  2  x  7619.  33.  6  x  3748. 

10.  4  x  4968.      22.  3  x  9167.  34.  7  X  4873. 

11.  9x9786.               23.  4x4682.  35.  8x5329. 

12.  7x3715.               24.  5x2864.  36.  9x9235. 

Ex.  39. 

Multiply  by  2 ;  by  3  ;  and  so  on  to  9 : 

1.  2739.             4.  7658.              7.  7463.  10.  6483. 

2.  4519.             5.  5396.              8.  8367.  11.  3526. 

3.  8526.             6.  5783.              9.  8562.  12.  5417. 

Multiply  by  20 ;  by  30 ;  and  so  on  to  90  : 

13.  5732.           14.  6749.            15.  8345.  16.  7952. 

Multiply  by  200 ;  by  300 ;  and  so  on  to  900 : 

17.  6738.           13.  3579.            19.  5742.  20.  5793. 

Multiply  by  2000;  by  3000;  and  so  on  to  9000: 

21.   4827.            22.  9357.             23.  6519.  24.  7953. 


54  MULTIPLICATION. 


58,  Suppose  the  product  of  649  X  4587  is  required.     The 
multiplier  649  is  600  +  40  +  9,  and  the  product  is  found 
by  multiplying  by  9,  then  by  40,  and  then  by  600,  and 
adding  the  partial  products.     Thus, 

4587 
649 

9  times  the  multiplicand  =      41283  ) 
40  times  the  multiplicand  =    183480  I    Fartl; 
600  times  the  multiplicand  =  2752200  j  Products- 

649  times  the  multiplicand  =  2976963 

59,  The  zeros  at  the  right  of  the  partial  products  do  not 
affect  the  result  of  the  addition,  and  may  be  omitted  if  care 
is  taken  to  put  the  right-hand  digit  of  each  partial  product 
directly  under  the  multiplier  used.     Thus, 

4587 
649 
41283 
18348 

27522 
2976963 

60,  If  the  multiplier  contains  zeros,  the  products  that 
correspond  to  them  will  be  zero,  and  need  not  be  written. 

Find  the  product  of  2007  X  4587. 


4587 

2007 
32109 

9174  Proof: 

9206109 


2007 
4587 
14049 
16056 
10035 
8028 


I  9206109 

61.  To  test  the  accuracy  of  the  work  in  multiplication, 
interchange  the  multiplicand  and  the  multiplier.  If  the 
numerical  result  is  the  same  in  both  cases,  as  in  the  last 
example,  the  work  may  be  assumed  to  be  correct. 


MULTIPLICATION.  55 


Ex.  40. 

Find  the  products  of: 

1.  27x8436.  13.  83x8495. 

2.  26  X  7358.  14.  86  x  5283. 

3.  36x3579.  15.  91x5246. 

4.  37  X  5684.  16.  93  x  6475. 

5.  45x5823.  17.  26x8167. 

6.  43  x  4263.  18.  29  x  7384. 

7.  53x4271.  19.  38x7496. 

8.  54  x  7538.  20.  34  x  4976. 

9.  64x9057.  21.  47x4982. 

10.  65x8154.  22.  46x8217. 

11.  78x6381.  23.  56x6284. 

12.  74x9472.  24.  57x9582. 

Ex.  41. 

Find  the  products  of : 

1.  364x6492.  13.  843x6527. 

2.  327x4756.  14.  935x5729. 

3.  283  x  5718.  15.  297  X  7186. 

4.  465x3862.  16.  487x8526. 

5.  592x4718.  17.  752x3849. 

6.  583x5926.  18.  594x6392. 

7.  647x8529.  19.  265x6973. 

8.  637  x  6548.  20.  378  x  7495. 

9.  741  x  9438.  21.  374  x  8247. 

10.  758x4857.  22.  648x9238. 

11.  824x3741.  23.  864x9753. 

12.  826x3297.  24.  798x5937. 


56  MULTIPLICATION. 


Ex.  42. 

1.  What  will  29  acres  of  land  cost  at  $475  an  acre? 

2.  What  will  89  passenger  cars  cost  at  $3785  a  car? 

3.  A  square  mile  contains  640  acres.     How  many  acres 

in  a  county  containing  936  square  miles? 

4.  If  a  cotton  factory  makes  9660  yards  of  cloth  daily, 

how  many  yards  will  the  factory  make  in  a  year 
(313  days)? 

5.  The  cost  of  building  a  certain  road  was,  on  the  aver- 

age, $  1789  a  mile.  What  was  the  cost  of  327  miles 
of  this  road  ? 

6.  If  a  field  contains   2340  hills   of  potatoes,  and   the 

average  number  of  potatoes  in  a  hill  is  12,  how 
many  potatoes  are  there  in  the  field  ? 

7.  If  a  saw  mill  turns  out  5708  feet  of  boards  in  a  day, 

how  many  feet  will  it  turn  out  in  294  days? 

8.  A  pound  of  platinum  is  worth  $85.     If  4730  pounds 

are  obtained  yearly  from  South  America  and  the 
Ural  Mountains,  what  is  the  value  'of  the  whole 
amount? 

9.  Two  cities  294  miles  apart  are  to  be  connected  by  a 

railroad,  at  a  cost  of  $24,645  per  mile.  What  will 
be  the  cost  of  the  road  ? 

10.  If  125  tons  of  steel  rails  are  required  for  one  mile  of 

railroad,  how  many  tons  will  be  necessary  for  389 
miles  ? 

11.  A  mile  contains  5280  feet.     How  many  feet  in  542 

miles  ? 


MULTIPLICATION.  57 


12.  The  garrison  of  a  fort  consumes  785  pounds  of  bread 

a  day.  How  many  pounds  will  be  consumed  in  3 
years  of  365  days  ? 

13.  If  a  railway  train  runs  38  miles  in  an  hour,  how  many 

miles  will  it  run  in  84  trips  of  3  hours  each  ? 

14.  A  square  mile  contains  640  acres.     How  many  acres 

are  there  in  3481  square  miles  ? 

15.  At  the  rate  of  1275  words  in   an  hour,  how  many 

words  can  be  sent  over  a  telegraph  line  in  108 
hours? 

16.  A  clock  strikes  156  times  a  day.     How  many  times 

does  it  strike  in  a  leap  year  (366  days)  ? 

17.  If  a  swallow  destroys  daily  500  insects,  how  many  will 

it  destroy  in  92  days  ? 

18.  At  27  bushels  an  acre,  how  many  bushels  of  wheat 

will  be  harvested  from  640  acres  ? 

19.  A  good  cow  yields  168  pounds  of  butter  a  year.     If  it 

takes  215,000  cows  to  supply  London  with  butter, 
how  many  pounds  of  butter  are  consumed  in  that 
city  annually  ? 

20.  If  sound  travels  at  the  rate  of  1120  feet  in  a  second, 

how  many  feet  distant  is  a  cloud  where  the  thunder 
clap  follows  the  flash  of  lightning  in  9  seconds  ? 

21.  Find  the  weight  in  pounds  of  5792  iron  bars,  each 

weighing  24  pounds. 

22.  From  what  number  can  847  be  subtracted  307  times, 

and  leave  a  remainder  of  49  ? 

23.  If  19  men  can  do  a  piece  of  work  in  31  days,  how 

many  days  will  it  take  one  man  to  do  it  ? 


58  MULTIPLICATION. 


24.  If  an  army  consists  of  24  regiments  averaging  913  men 

each,  how  many  men  are  there  in  the  whole  army  ? 

25.  If  one  acre  produces  211  pounds  of  cotton,  how  many 

pounds  will  933,000  acres  produce  ? 

26.  If  one  acre  produces    154   pounds   of  tobacco,  how 

many  pounds  will  10,070  acres  produce  ? 

27.  If  one  acre  produces  28  bushels  of  oats,  how  many 

bushels  will  911,200  acres  produce? 

28.  If  one  acre  produces  42  bushels  of  corn,  how  many 

bushels  will  201,106  acres  produce? 

29.  If  one  acre  produces  17  bushels  of  wheat,  how  many 

bushels  will  613,263  acres  produce? 

30.  If  one  acre  produces  227  bushels  of  potatoes,  how 

many  bushels  will  19,121  acres  produce? 

31.  If  one  acre  produces  23  bushels  of  barley,  how  many 

bushels  will  237,769  acres  produce  ? 

32.  If  one  acre  produces  19  bushels  of  winter  rye,  how 

many  bushels  will  27,119  acres  produce? 

33.  British  India  has  a  population  of  150  to  the  square 

mile,  and  contains  1,004,616  square  miles.    Find  its 
population. 


CHAPTER  V. 

DIVISION. 

62,  To  divide  $42  by  6  is  to  find  the  number  of  dollars 
that  must  be  taken  6  times  to  make  $42.     Again,  to  divide 
$42  by  $6  is  to  find  the  number  of  times  that  it  is  neces- 
sary to  take  $6  to  make  $42.     In  either  case,  the  product 
and  one  factor  are  given  and  the  other  factor  is  required. 
Hence, 

63,  Division  is  an  operation  by  which  when  the  product 
and  one  factor  are  given  the  other  factor  is  found. 

64,  The  number  to  be  divided  is  called  the  dividend,  the 
number  by  which  the  dividend  is  to  be  divided  is  called 
the  divisor,  arid  the  result  is  called  the  quotient, 

65,  Division  is  indicated  by  the  sign  of  division  -s-,  or  by 
writing  the  dividend  over  the  divisor  with  a  line  between 
them.    Thus,  each  of  the  expressions  42-*- 6  =  7,  and  ^=7, 
means  and  is  read  "  forty- two  divided  by  six  equals  seven." 

Ex.  43.     (Oral) 

2x8-  .-.16-5-2  =  2x6=  .-.  12-*- 6  = 

16-*- 8=  12-*- 2  = 

2x2-  .-.  4-*- 2=  2x3-  .•.6*1-8  = 

6-s-2  = 

2x5-  .-.10-5-2=:;  2x7=  .-.14-5-7  = 

10-5-5  =  14-i-2  = 


60  DIVISION. 


2x9  = 

.-.18-4-2  = 

5x9  = 

.-.45  +  5  = 

18  +  9  = 

45  +  9  = 

3x4  = 

.-.12  +  4  = 

5x8  = 

.-.40  +  8  = 

12  -4-  3  = 

40  +  5  = 

3x3  = 

.*.  9  +  3- 

5X3  = 

.-.15  +  3  = 

15  +  5  = 

3x6  = 

.•.18-4-6  = 

5x6  = 

.-.30  +  6  = 

18  +  8  = 

30  +  5  = 

3x9  = 

.•.27-9  = 

5x4  = 

.-.20  +  4  = 

27  +  3  = 

20  +  5  = 

3x7  = 

/.21  +  3  = 

5x7  = 

.-.35+7  = 

21  +  7  = 

35  +  5  = 

3x8  = 

.-.24  +  8  = 

6x9  = 

.-.54  +  6  = 

24  +  3  = 

54  +  9  = 

3x5  = 

.-.15  +  5  = 

6x3  = 

.-.18  +  6  = 

15  +  3  = 

18  +  3  = 

4x5==. 

.-.20-4  = 

6x6  = 

.-.36  +  6  = 

20  +  5  = 

4x3  = 

/.12  +  4  = 

6x7  = 

.-.42  +  6  = 

12  +  3  = 

42  +  7  = 

4x6  = 

.-.24  +  6  = 

6x8  = 

.-.48  +  8  = 

24  +  4  = 

*  48  +  6  = 

4x9  = 

.-.36  +  9  = 

6x5  = 

.-.30  +  6  = 

36  +  4  = 

30  +  5  = 

4x7  = 

.-.28+7  = 

6x4  = 

.-.24  +  6  = 

28  +  4  = 

24  +  4  = 

4x8  = 

.-.32  +  4  = 

7x3  = 

.-.21  +  3  = 

32  +  8  = 

21  +  7  = 

4x4  = 

.-.16  +  4  = 

7x9  = 

.-.63  +  9  = 

63  +  7  = 

5x5  = 

.-.25-5  = 

7x7  = 

.-.49  +  7  = 

DIVISION. 


7x4  = 

.-.28-4-4  = 

8x9  = 

.-.724-9  = 

28-4-7  = 

72-4-8  = 

7x8  = 

.-.56-4-7  = 

8x4  = 

.-.32-4-4  = 

56-4-8  = 

32-4-8  = 

7x5  = 

.-.35-f-7  = 

9x3  = 

.-.274-9  = 

35-4-5  = 

27  -*-  3  = 

7x6  = 

.-.42-^-6  = 

9x5  = 

.-.45-4-9  = 

42-4-7  = 

45-4-5  = 

8x8  = 

.-.64-5-  8  = 

9x9  = 

.-.81  4-  9  = 

8x3  = 

.  -.24-5-8  = 

9x6  = 

.-.54^-9  = 

24  -v-  3  = 

54  -4-  6  = 

8x7  = 

.-.56-4-7  = 

9x8  = 

.-.72-8  = 

56-^-8  = 

72-4-9  = 

8x5  = 

.•.40-4-5  = 

9x7  = 

.-.63-4-9  = 

40-^-8  = 

83-4-7  = 

66.  In  the  following  exercises,  the  divisor  for  each  line  of 
dividends  is  written  at  the  left.  The  quotients  should  be 
named  without  a  moment's  hesitation. 

Ex.  44.  (Om/.) 

1.  6)54184212_62436603048 

2.  4)281620324036_42412_8 

3.  8)32486416402456_872_0 

4.  9)_0277281186354_94536 

5.  7)  21      _0      14      _7      28      42      35      56      63      49 

6.  5)304520^405025153510 

7.  3)    9      12        3      30      27      18      15      21      _6      24 


62  DIVISION. 


Ex.  45.     (Oral.) 

Give   the   quotients   and   remainders   in   the   following 
examples : 

1.2)11  _I_91^i31418191617 
2.  3)j7111013'l6141715^519 
3.4)19  _71317251522313329 

4.  9)71831525341719622644 

5.  7)15      192738405448601739 

6.  8)23143117_92537687128 

7.  6)20151927321013455740 

8.  5)14_91321431249322938 
9.7)11      182637345347591633 

10.  5)13112216423248314937 
11.6)21  163119112639465641 
12.  8)33343918274169707563 
13.9)73  168429354351648070 
14.5)37  413427362328334448 
15.6)37  4417105158253459  50 
16.  8)26393042532036435157 
17.9)21  372341471155506065 
18.7)13  236146556925586218 
19.  8)55677344615074655277 


DIVISION.  63 


SHORT  DIVISION. 

67.  When  the  divisor  is  so  small  that  the  work  can  be 
performed  mentally,  the  process  is  called  Short  Division,  and 
will  be  understood  from  the  following  examples : 

(1)  Divide  697,425  by  3. 

The  divisor  is  written  at  the  left  of  the  dividend,  as  in  the  margin. 
Wording.  3  in  6,  2 ;  in  9,  3 ;  in  7,  2 ;  in  14,  4 ;  in 
3)697425  22,  7  ;  in  15,  5. 

232475  Here  the  divisor  is  contained  in  6  twice,  in  9  three 
times,  and  in  7  twice  with  remainder  1 ;  this  1  is  equal 
to  10  of  the  next  lower  order,  and  with  the  4,  the  next  order  of  the 
dividend,  makes  14.  Then  14  is  divided  by  3  ;  the  quotient  is  4  with 
remainder  2;  this  2  is  equal  to  20  of  the  next  lower  order,  and  with 
the  2  makes  22.  Then  22  is  divided  by  3 ;  the  quotient  is  7  with 
remainder  1.  Then  15  is  divided  by  3,  and  the  quotient  is  5. 

(2)  Divide  4,236,158  by  7. 

7)4236158 

605165  with  remainder  3. 

In  this  example,  7  is  not  contained  in  3,  BO  0  is  the  second  figure 
of  the  quotient:  then  the  next  figure  6  of  the  dividend  is  joined  to 
the  3,  making  36,  and  the  division  is  continued.  When  the  division 
is  finished,  there  is  a  remainder  3. 

(3)  Divide  54,123  by  9. 

9)54123 

6013  with  remainder  6. 

Each  quotient  figure  is  of  the  same  order  of  units  as  the  right-hand 
figure  of  that  part  of  the  dividend  used  in  obtaining  it.  Thus,  54  in 
this  example  are  54  thousands,  and  the  first  figure  of  the  quotient  ia 
6  thousands. 

(4)  Divide  $23,087  by  5. 

5)  $23087 

$4617  with  $2  remaining. 


64  DIVISION. 


In  this  example,  we  are  required  to  divide  23087  dollars  into  five 
equal  parts,  and  find  the  number  of  dollars  in  each  part.  The  answer 
is  4617  dollars,  with  2  dollars  over.  The  complete  quotient  may  be 
written  $4617$. 

(5)   Divide  $23,087  by  $5. 

$5)  $23087 

4617  with  $  2  remaining. 

In  this  example,  we  are  required  to  find  the  number  of  times  we 
can  take  away  $5  from  $23,087,  and  the  answer  is  4617  times,  with 
$2  left  over.  The  complete  quotient  may  be  written  4617$;  and 
the  meaning  is,  that  we  can  take  $5  away  4617  times  from  $23,087, 
and  the  next  time  have  $  2  to  take  away. 

68,  The  last  two  examples  illustrate  the  different  mean- 
ings of  division.     When   the  divisor  corresponds  to  the 
multiplier  in  multiplication  the  quotient  corresponds  to  the 
multiplicand,  and  denotes  the  same  kind  of  units  as  the  divi- 
dend;  when  the  divisor  corresponds  to  the  multiplicand 
the  quotient  corresponds  to  the  multiplier,  and  denotes  the 
number  of  times  the  divisor  must  be  taken  to  obtain  a 
quantity  equal  to  the  dividend. 

69.  A  number,  when  divided  by  10,  will  have  a  quotient 
consisting  of  the  same  series  of  figures,  the  last  one  being 
cut  off  for  the  remainder.     Thus,  35764-^10  =  3576  with 
remainder  4.     In  this  case,  the  value  of  each  figure  in  the 
result  is  diminished  ten-fold,  the  tens  becoming  units,  the 
hundreds  becoming  tens,   and  so  on.      A   number,  when 
divided  by  100,  1000,  etc.,  will  have  the  same  series  of 
figures  in  the  quotient,  the  last  two,  three,  etc.,  figures  being 
cut  off  for  the  remainder.     Hence, 

When  a  divisor  ends  in  one. or  more  zeros,  cut  off  the 
zeros  and  an  equal  number  of  figures  from  the  right  of 
the  dividend,  perform  the  division  with  the  numbers  left, 


DIVISION.  65 


and  for  the  total  remainder  annex  the  figures  cut  off  from 
the  dividend  to  the  remainder  from  the  division. 

Divide  5,786,342  by  200. 

200)57863^2 

28931  with  remainder  142. 

In  this  example,  we  cut  off  the  two  zeros  at  the  right  of  the  divisor 
and  two  figures  at  the  right  of  the  dividend ;  then  we  divide,  putting 
the  first  figure  of  the  quotient  under  the  figure  8,  which  is  the  right- 
hand  figure  of  the  first  partial  dividend  when  the  entire  divisor  200 
is  used. 

70,  The  product  of  the  divisor  and  quotient  increased  by 
the  remainder  is  equal  to  the  dividend.  Hence, 

To  test  the  accuracy  of  the  work  of  division,  find  the 
product  of  the  divisor  and  quotient,  and  to  this  product 
add  the  remainder ;  this  result  will  be  equal  to  the  divi- 
dend if  the  work  is  correct. 

Thus,  in  the  last  example, 

200  X  28,931  -  5,786,200, 
and  5,786,200  +  142  =  5,786,342  (the  dividend). 

Ex.  46. 

Find  the  quotients  of : 

1.  48-*- 2.        10.  75 +-5.        19.  91-*- 8.         28.  815-*- 5. 


2. 

72 

-*-8. 

11. 

98+- 

7. 

20. 

94+- 

9. 

29. 

714 

+-6. 

3. 

56 

+-4. 

12. 

92+- 

4. 

21. 

94+- 

5. 

30. 

826 

+-7. 

4. 

85 

•*-5, 

13. 

57+- 

2. 

22. 

87+- 

4. 

31. 

952 

+-8. 

5. 

96 

-f-6. 

14. 

83+- 

3. 

23. 

95+- 

6. 

32. 

972 

+-9. 

6. 

84 

+-7. 

15. 

75+- 

4. 

24. 

77+- 

3. 

33. 

912 

+-8. 

7. 

96 

f-8. 

16. 

48+- 

5. 

25. 

734  H 

-2. 

34. 

492 

+-4. 

8. 

99 

+-9. 

17. 

77+- 

6. 

26. 

768  H 

h-a 

35. 

675 

+-5. 

9. 

90 

-1-8. 

18. 

82+- 

7. 

27. 

956  H 

n4. 

36. 

918 

+  6. 

66  DIVISION. 


37. 

513-4- 

2. 

53. 

9354-4- 

6. 

69. 

4017 

.§_ 

7. 

38. 

719-4- 

3. 

54. 

8176  -4- 

7. 

70. 

7139 

-4- 

8. 

39. 

623^- 

4. 

55. 

9456-4- 

8. 

71. 

9415 

-4- 

6. 

40. 

749^- 

5. 

56. 

8568-4- 

9. 

72. 

8793 

-4- 

5. 

41. 

875-r- 

6. 

57. 

3712-4- 

8. 

73. 

3794 

•4- 

2. 

42. 

643-^ 

7. 

58. 

2226-4- 

7. 

74. 

7929 

-4- 

3. 

43. 

927-4- 

8. 

59. 

2550-4- 

6. 

75. 

6728 

-4- 

4. 

44. 

705-4- 

9. 

60. 

2895-4- 

5. 

76. 

6380 

-4- 

5. 

45. 

591-*- 

8. 

61. 

5391^- 

o. 

77. 

8322 

-4- 

6. 

46. 

853-4- 

7. 

62. 

7418-4- 

3. 

78. 

9219 

^ 

7. 

47. 

735^- 

6. 

63. 

5327-4- 

4. 

79. 

7395 

^~ 

2. 

48. 

923-4- 

5. 

64. 

8236-4- 

5. 

80. 

7684 

-J- 

3. 

49. 

7594-^-2. 

65. 

7129-4- 

6. 

81. 

7315 

-s- 

4. 

50. 

7458  H 

K& 

66. 

8513+- 

7. 

82. 

8369 

-4- 

5. 

51. 

9656  -4-  4. 

67. 

9237^- 

8. 

83. 

5869  --  6. 

52. 

7985  H 

h5. 

68. 

5682  -4- 

9. 

84. 

4239 

+ 

7. 

Divide  by  2 ;  by  3  ;  and  so  on  to  9  : 
85.  5794.    86.  4572.    87.  9785.    88.  7163. 

Divide  by  20  ;  by  30 ;  and  so  on  to  90. 
89.  8239.    90.  5197.    91.  3274.    92.  5834. 

Divide  by  200 ;  by  300  ;  and  so  on  to  900  : 
93.  4571.    94.  5768.    95.  9563.    96.  9876, 

Ex.  47. 

I.  There  were  72  children  in  a  Sunday-school,  and  they 
walked  two  and  two  to  church.  How  many  rows 
would  they  make  ?  How  many  rows  would  there 
have  been  if  they  had  walked  three  and  three  ? 


DIVISION.  67 


2.  A  boy  had  97  filberts.     He  kept  34  for  himself,  and 

divided  the  rest  equally  among  his  9  class-mates. 
How  many  did  he  give  to  each  ? 

3.  How  many  times  must  we  take  the  number  7  to  make 

819  ?     How  many  times  the  number  9  ? 

4.  Divide  a  paper  of  264  pins  equally  into  8  papers. 

5.  2691   poles  were  used  in  a  certain  hop-yard,  and  3 

were  required  for  each  plant.      How  many  plants 
were  there  ? 

6.  A  blacksmith  uses  7  nails  in  putting  on  one  shoe,  and 

in  one  day  he  used  336  nails.    How  many  hoofs  did 
he  shoe  ? 

7.  A  forest  of  1995  trees  is  to  be  thinned  by  cutting  down 

1  tree  in  7.     How  many  will  be  taken  out  ? 

8.  A  regiment  consists  of  1200  men  and  60  officers.    How 

many  men  are  there  to  each  officer  ? 

9.  When  beef  is  $7  per  hundred-weight,  how  many  hun- 

dred-weight can  be  bought  for  $9,700,327? 

10.  How  many  tons  of  coal,  at  $9,  can  be  bought  for 

$3,596,301? 

11.  A  wagon  travels  58,068  feet.     How  many  times  will 

a  wheel  12  feet  in  circumference  turn  in  going  that 
distance  ? 

12.  A  square  yard  contains 9  square  feet.   How  many  square 

yards  in  3,917,502  square  feet? 

13.  Aaron  Reed  left  $325,645  for  his  wife  and  four  chil- 

dren.    How  much  had  each,  if  the  property  was 
divided  equally  among  them  ? 


68  DIVISION. 


14.  A  grocer  sells  brown  sugar  at  $9  per  hundred- weight. 

If  he  receives  $976,482,  how  many  hundred-weight 
does  he  sell  ? 

15.  John  Brown  paid  $375,008  for  a  tract  of  wild  land,  at 

$8  per  acre.     How  many  acres  did  he  buy? 

16.  How  many  tons  of  coal,  at  $7  per  ton,  can  be  pur- 

chased for  $3,785,908? 

17.  A  merchant  received  $397,640  in  selling  a  quantity 

of  flour,  at  $8  per  barrel.     How  many  barrels  did 
he  sell  ? 

18.  What  must  be  paid  for  12  yards  of  cloth,  if  5  yards 

cost  $25? 

SOLUTION.  If  5  yards  cost  $25,  to  find  the  cost  of  1  yard 
$25  must  be  divided  by  5;  $25-*- 5  =  $5,  cost  of  1  yard.  12 
yards  will  cost  12  x  $5  =  $60.  Ans. 

19.  A  drover  paid  $20  for  5  sheep.    What  will  be  the  cost 

of  125  sheep? 

20.  Three  cows  cost  $156.     What  must  be  paid  for  27 

cows? 

21.  If  7  tons  of  hay  cost  $105,  what  will  be  the  cost  of  63 

tons? 

22.  If  9  barrels  of  flour  are  worth  $63,  how  many  barrels 

of  apples,  at  $3  a  barrel,  will  pay  for  72  barrels  of 
flour? 

23.  If  7  cords  of  birch  wood  are  worth  $28,  how  many 

cords  of  birch  wood  will  pay  for  6  barrels  of  sugar 
worth  $16  a  barrel? 

24.  If  12  men  do  a  piece  of  work  in  12  hours,  how  many 

hours  would  it  take  8  men  to  do  the  same  work  ? 


DIVISION 


69 


LONG  DIVISION. 

71,  The  process  of  Long  Division  is  the  same  as  that  of 
Short  Division,  except  that  the  work  is  written  in  full,  and 
the  quotient  is  written  over  the  dividend. 

Divide  41,668  by  78. 

The  beginner  will  find  it  convenient  to  form  a  table  of 
products  of  the  divisor  by  the  numbers  1,  2,  3, ,  as  follows : 


1  X  78  =    78 

4x78-312 

7  x  78  -  546 

2  x  78  =  156 

5  X  78  =  390 

8  X  78  =  624 

3  x  78  =  234 

6  x  78  -  468 

9  x  78  -  702 

The  third  product  is  found  by  adding  the  first  and  sec- 
ond products,  the  fourth  by  adding  the  first  and  third,  and 
so  on. 

As  78  is  more  than  41,  it  is  necessary  to  take  three  figures  of  the 
dividend  for  the  first  partial  dividend.  Of  the  products  in  the  table 
that  do  not  exceed  419  the  greatest  is  390, 
that  is,  5  X  78.  Hence  the  first  quotient 
figure  is  5,  and  is  written  over  the  9  in  the 
dividend ;  then  390  is  subtracted  from  419. 
To  the  remainder  29,  the  next  figure  9  of 
the  dividend  is  annexed.  Of  the  products 
that  do  not  exceed  299,  the  greatest  is  234, 
that  is,  3  x  78.  Hence  3  is  the  next  figure 
of  the  quotient,  and  the  next  remainder  is 
65,  to  which  the  8  of  the  dividend  is  an- 
nexed. Of  the  products  that  do  not  exceed 
658,  the  greatest  is  624,  that  is,  8  X  78.  Hence  the  next  figure  of  the 
quotient  is  8,  and  the  remainder  34. 

After  a  little  practice  the  operation  of  division  can  be 
performed  without  the  aid  of  a  table  of  products.  Each 
quotient  figure  is  estimated  by  taking  for  a  trial  divisor  the 
left-hand  figure  of  the  divisor  (or  the  left-hand  figure  in- 


OPERATION. 
538 

78)41998 
390 
299 
234 
658 
624 

34  remainder. 


70  DIVISION. 


creased  by  1,  when  the  next  figure  is  greater  than  5),  and 
by  taking  for  a  trial  dividend  one  or  two  figures  only  of 
each  partial  dividend.  When  the  trial  divisor  is  increased 
by  1,  the  trial  dividend  should  be  increased  by  1. 

Divide  2,791,163  by  394. 

The  first  partial  dividend  is  2791.     As  9,  the  second  figure  of  the 
divisor,  is  greater  than  5,  we  take  4  for  a  .trial  divisor.     As  we  have 
increased  the  trial  divisor,  we  increase 
the  trial  dividend  by  1,  making  it  28. 
4  is  contained  in  28  7  times.     We  write  7084 

the  7  over  the  1,  and  multiply  the  divi-     394)  2791163 
Bor  394  by  7.     We  subtract  the  product  2758 

2758  from  2791  and  have  for  a  remain-  3316 

der  33,  to  which  we  annex  the  1  of  the  3152 

dividend.    As  331  is  less  than  394,  the  1643 

next  quotient  figure  is  0.     To  331  we  1576 

annex  the  next  figure  6  of  the  dividend.  £7  remainder. 

4  is  contained  in  34  8  times.  We  there- 
fore write  8  for  the  next  quotient  figure,  and  find  the  product  of 
8  X  394  to  be  3152.  The  remainder  obtained  by  subtracting  3152  is 
164,  to  which  the  3  of  the  dividend  is  annexed.  4  is  contained  4 
times  in  17.  The  product  of  4  x  394  is  1576,  and  this  subtracted 
from  1643  leaves  67  for  the  final  remainder. 

NOTE.  If  the  product  of  the  divisor  by  the  quotient  figure  is 
greater  than  the  partial  dividend,  the  quotient  figure  is  too  large, 
and  must  be  diminished ;  and,  if  the  difference  between  the  partial 
dividend  and  the  product  of  the  divisor  by  the  quotient  figure  is 
greater  than  the  divisor,  the  quotient  figure  is  too  small  and  must  be 
increased. 

Ex.  4a 

Find  the  quotients  of : 

1.  4386-^21.              5.  9357 -f- 61.  9.  6985-^22. 

2.  5271-^-31.              6.  5263 -f- 71.  10.  9876 -f- 32. 

3.  8056-^-41.              7.  3046 -f- 82.  11.  2378-^42. 

4.  7158-^51.              8.  7219 -J- 92.  12.  4068-^52. 


DIVISION. 


71 


13.    8359-5-63. 

21.      6,543-5-68. 

29.    79,853-4-63 

14.   4573-73. 

22.      8,319  -4-  78. 

30.   82,569-4-73. 

15.    7358-4-84. 

23.     5,432-4-89. 

31.    94,365-5-84. 

16.   3985-94. 

24.      9,753-4-99. 

32.    98,765-94. 

17.    6973-4-25. 

25.   41,268-4-21. 

33.   82,639-f-25. 

18.    7413-36. 

26.    74,306-4-31. 

34.    64,372-5-35. 

19.   8765-4-47. 

27.    89,415  -i-  42. 

35.   59,036-5-46. 

20.    7654-4-57. 

28.    67,834  -*-  52. 

36.   42,837-4-56. 

Ex.  49. 

Find  the  quotients  of  : 

1.  84,317  -f-  67. 

13.  437,650-^-23. 

25.  437,650-4-53. 

2.  72,659-4-77. 

14.  657,320-35. 

26.  657,320  -*-  65. 

3.  64,980-4-88. 

15.  327,045-47. 

27.  327,045  -*-  77. 

4.  52,196  -f-  98. 

16.  632,008-^-59. 

28.  632,008-5-89. 

5.  47,028  -f-  29. 

17.  437,650-^-33. 

29.  437,650-4-63. 

6.  74,369-39. 

18.  657,320-^-45. 

30.  657,320-4-75. 

7.  54,371-4-14. 

19.  327,045-57. 

31.  327,045-4-87. 

8.  68,594-4-15. 

20.  632,008  -i-  69. 

32.  632,008-4-99. 

9.  73,109-4-16. 

21.  437,650-^-43. 

33.  43*7,650-4-73. 

10.  82,563-4-17. 

22.  657,320-5-55. 

34.  657,320-85. 

11.  94,069  -f-  18. 

23.  327,045  -i-  67. 

35.  327,045-5-97. 

12.  47,938-5-19. 

24.  632,008-^-79. 

36.  632,008-4-29. 

Ex.  50. 

Find  the  quotients  of  : 

1.  50,576-5-101. 

7.   76,593-^-415. 

13.  96,432-4-781. 

2.  50,576  -f-  102. 

8.  76,593-4-516. 

14.  96,432-4-592. 

3.  50,576-4-203. 

9.  76,593-^-621. 

15.  96,432^-864. 

4.  50,576-5-205. 

10.  76,593-4-732. 

16.  96,432-5-972. 

5.  50,576-4-302. 

11.  76,593-4-843. 

17.  96,432  +  492. 

6.  50,576-^-106. 

12.  76,593-5-954. 

18.  96.432-5-993. 

72  DIVISION. 


Ex.  51. 

Find  the  quotients  of : 

1.  861,345-^-4001.  7.  730,604-8403. 

2.  861,345-2048.  8.  972,817-^-7184. 

3.  861,345-^3507.  9.  854,235-8794. 

4.  861,345-^-6409.  10.  730,604-^-5748. 

5.  861,345-8157.  11.  972,817-4981. 

6.  861,345-^-3965.  12.  730,604-1984. 

Ex.  52. 

1.  How  many  stoves  can  be  bought  for  $1120,  if  one 

stove  costs  $35? 

2.  If  a  carriage  is  valued  at  $  144,  how  many  carriages 

can  be  bought  at  the  same  rate  for  $54,000? 

3.  A  horse  dealer  bought  a  horse  for  $125.     How  many 

horses  could  he  buy  for  $60,625,  at  the  same  rate? 

4.  How  many  barrels  of  sugar  can  be  bought  for  $8352 

when  $36  is  paid  for  one  barrel? 

5.  A  merchant  sold  297  barrels  of  flour  for  $2673.    How 

much  did  he  get  a  barrel  ? 

6.  George  Clifford  paid  $10,250  for  oxen,  paying  on  the 

average  $82  an  ox.     How  many  did  he  buy? 

7.  How  many  days  will  it  take  a  man  to  dig  a  ditch  864 

feet  long,  if  he  can  dig  48  feet  a  day  ? 

8.  One  share  of  a  certain  bank  stock  is  worth  $98.    How 

many  shares  can  be  bought  for  $  22,050  ? 


DIVISION.  73 


9.    A  farmer  sold  19  sheep  for  $152.     For  how  much  a 
head  did  he  sell  them  ? 

10.  John  Jones  paid  $1752  for  lambs  at  an  average  price 

of  $4.     How  many  did  he  buy  ? 

11.  A  fruit  grower  received  $1755  for  195  barrels  of  cran- 

berries.     What  was  the  price  per  barrel? 

12.  In  one  square  foot  there  are  144  square  inches.     How 

many  square  feet  in  1,375,920  square  inches? 

13.  A  public  library  has  a  yearly  circulation  of  56,966 

books.     How  many  books  are  taken  daily,  if  the 
library  is  open  313  days  in  a  year  ? 

14.  One   mile   contains  320  rods.     How  many  miles  in 

348,160  rods  ? 

15.  If  the  dividend  is  514,478,  the  divisor  327,  and  the 

remainder  107,  what  is  the  quotient  ? 

16.  A  railroad  478  miles  in  length  cost  $3,500,872.    What 

was  the  average  cost  per  mile  ? 

17.  How  many  house  lots,  at  $321  for  each,  can  be  bought 

for  $772,326? 

18.  A  company  of  547  men  took  equal  shares  in  a  mine 

valued  at  $705,083.     How  much  money  did  each 
man  invest? 

19.  If  325  workmen  are  paid  $583,700,  what  sum  does 

each  receive  ? 

20.  At  $89  per  acre,  how  many  acres  of  land  can  be  pur- 

chased for  $713,513? 

21.  Divide  one  million  three  hundred  seventy-five  thou- 

sand eight  hundred  nine  by  two  hundred  eighty- 
seven. 


74  DIVISION. 


22.  A  ship  averaging  215  miles  per  day  has  to  sail  3678 

miles.  How  many  days  will  be  required  for  the 
trip? 

23.  A  New  Orleans  merchant  sends  to  New  York  376,705 

gallons  of  molasses.  How  many  casks  will  there  be 
if  each  cask  contains  235  gallons  ? 

24.  The   capital   and   surplus   of  a   bank   amounting   to 

$518,077  belonged  to  679  stockholders.  What  is 
the  average  amount  belonging  to  each  stockholder  ? 

25.  If  34,823  tons  of  coal  are  required  for  97  steamships, 

what  is  the  average  number  of  tons  for  each  ? 

26.  A  carpet  factory  running  45  looms  makes  17,820  yards 

of  carpet  in  a  fortnight.  What  number  of  yards  is 
woven  by  each  loom  on  the  average  ? 

27.  A  cotton  planter  raises  428,243  pounds  of  cotton.     If 

the  cotton  is  put  into  bales,  weighing  on  the  average 
401  pounds,  what  will  be  the  whole  number  of 
bales? 

28.  In  one  cubic  foot  there  are  1728  cubic  inches.     How 

many  cubic  feet  are  there  in  a  pile  of  wood  contain- 
ing 3,507,840  cubic  inches? 

29.  In  how  many  hours  will  a  cistern  holding  3330  gal- 

lons be  filled  by  a  pipe  that  discharges  into  it  185 
gallons  an  hour  ? 

30.  An  army  officer  paid  $107  for  a  horse.     At  that  rate 

how  many  horses  can  he  buy  for  $317,897  ? 

31.  A  man  having  an  income  of  $3874  a  year  (52  weeks) 

spent  $2314  and  saved  the  rest.  How  much  did  he 
save  per  week  on  the  average? 


DIVISION.  75 


32.  What  number  subtracted  88  times  from  80,005  will 

leave  13  as  a  remainder  ? 

33.  How  many  rolls  of  carpet  at  $75  a  roll  can  be  bought 

for  $1275? 

34.  A  has  425  horses  valued  at  $58,650  ;  B  has  382  acres 

of  land  worth  $48,514.  What  is  the  difference  in 
value  between  one  of  A's  horses  and  an  acre  of  B's 
land? 

35.  If  the  dividend  is  325,682,  the  divisor  284,  and  the 

remainder  218,  what  is  the  quotient  ? 

36.  What  is  the  nearest  number  to  7196  that  will  contain 

372  without  a  remainder  ? 

37.  New  York  contains  47,620  square  miles,  Texas  262,290. 

How  many  states  as  large  as  New  York  can  be  made 
out  of  Texas,  and  how  many  square  miles  will  be 
left  over  ? 

38.  Dakota  contains  147,700  square  miles,  Massachusetts 

8040.  How  many  states  as  large  as  Massachusetts 
can  be  made  out  of  Dakota,  and  how  many  square 
miles  will  be  left  over  ? 

39.  In  1880  Texas  produced  550,872,000  pounds  of  cotton. 

Allowing  400  pounds  to  a  bale,  how  many  bales  of 
cotton  did  Texas  raise  that  year  ? 

40.  If  one  pound  of  sugar  is  obtained  from  18  sugar  canes, 

how  many  pounds  will  be  obtained  from  1,233,216 
canes  ? 


CHAPTER  VL 

DECIMALS. 

72,  Numbers  which  denote  whole  units  are  called  Integral 
numbers ;  but  it  is  often  necessary  to  express  parts  of  a 
unit. 

If  a  unit  is  divided  into  two  equal  parts,  each  part  is 
called  one-half,  and  is  expressed  by  £.  If  a  unit  is  divided 
into  three  equal  parts,  each  part  is  called  one-third,  and  is 
expressed  by  -J- ;  two  of  the  parts  are  called  two-thirds,  and 
are  expressed  by  -|.  Again,  if  a  unit  is  divided  into  four 
equal  parts,  each  part  is  called  one-fourth,  and  is  expressed 
by  \ ;  into  five  equal  parts,  each  part  is  called  one-fifth,  and 
is  expressed  by  •£ ;  into  six  equal  parts,  each  part  is  called 
one-sixth,  and  is  expressed  by  % ;  into  seven  equal  parts,  each 
part  is  called  one-seventh,  and  is  expressed  by  ^ ;  into  eight 
equal  parts,  each  part  is  called  one-eighth,  and  is  expressed 
by  % ;  into  nine  equal  parts,  each  part  is  called  one-ninth, 
and  is  expressed  by  -J-;  into  ten  equal  parts,  each  part  is 
called  one-tenth,  and  is  expressed  by  -fa. 

If  AB  (see  page  opposite)  represent  a  unit  of  length, 
each  division  of  the  line  next  below  AB  represents  one- 
half  of  a  unit ;  and  each  division  of  the  second  line  below 
AB  represents  one-third  of  a  unit ;  and  so  on. 

How  many  halves  of  a  unit  make  a  whole  unit  ? 

How  many  fourths  make  a  half?  how  many  make  a 
whole  unit? 

How  many  sixths  make  a  third?  how  many  make  a  half? 
how  many  make  a  whole  unit  ? 

How  many  eighths  make  a  half?  a  fourth?  a  whole  unit? 

How  many  tenths  make  a  fifth?  a  half?  a  whole  unit? 


DECIMALS.  77 


i 

i 

1 

1                        1                        1 

1 


i 


1  1              1 

1 

1             1 

i 

\ 

i 

i         i 

1 

1           1 

a         i 
i        i 

§ 

i 

l 

i 

1 
i 

i        3 

i          * 

1           1 

$        f 

i        i        i 

I 

\ 

§ 
i 

1 

i 

drAAdrAAAftfttt 

73,  When  a  unit  is  divided  into  ten  equal  parts,  and  we 
wish  to  express  in  figures  one  or  more  of  these  parts,  we  do 
not  usually  write  them  -fa,  y2^,  etc.,  but  we  write  1,  2,  3, 
etc.,  and  separate  the  number  which  denotes  parts  of  a  unit 
from  the  number  which  denotes  whole  units  by  a  decimal 
point.  Thus,  two  units  and  three-tenths  of  a  unit  are 
written,  i3. 

If  each  tenth  of  a  unit  is  divided  into  ten  equal  parts, 
that  is,  the  entire  unit  into  a  hundred  equal  parts,  each 
part  is  called  a  hundredth  of  the  unit ;  and  if  each  hun- 
dredth is  divided  into  ten  equal  parts,  that  is,  the  entire 
unit  into  a  thousand  equal  parts,  each  part  is  called  a  thou- 
sandth of  the  unit ;  and  so  on. 

These  tenth-parts  are  called  Decimal  parts,  from  the  Latin 
word  decem,  which  means  ten;  and  these  parts  are  com- 
monly called  Decimal  Fractions, 


78 


DECIMALS. 


Let  A B,  for  example,  represent  the  unit  of  length  by  which  a  certain 
distance  is  to  be  measured.  Suppose  the  given  distance  to  contain 
AS  137  times,  and  a  remainder  LM  to  be  left,  which 
is  less  than  AB.  Take  AC,  a  tenth  of  AB,  and 
suppose  AC  is  contained  in  LM ^  times,  with  a  re- 
mainder OM  less  than  AC.  Again,  suppose  AD,  a 
tenth  of  A 0  (that  is,  a  hundredth  of  AB),  to  be  con- 
tained in  OM  3  times,  with  a  remainder  less  than 
AD.  And  again,  suppose  a  tenth  of  AD  (that  is,  a 
thousandth  of  AB),  to  be  contained  in  this  last  re- 
mainder 9  times.  Then  the  whole  distance  expressed 
in  lengths  of  AB  will  be  137.439. 

The  series  of  figures  137.439  means  1  hun- 
dred +  3  tens  +  7  units  +  4  tenths  +  3  hun- 
dredths  +  9  thousandths  ;  as  1  hundred  =  10 
tens  =  100  units,  and  3  tens  =  30  units,  the 
integral  value  is  137  units;  so,  4  tenths =40 
hundredths  =  400  thousandths,  and  3  hun- 
dredths  =  30  thousandths;  the  decimal  value 
therefore  is  439  thousandths. 

If  the  unit  is  the  yard-stick,  the  whole  is 
read  "one  hundred  thirty-seven  and  four 
hundred  thirty-nine  thousandths  yards";  if 
the  unit  is  the  meter-stick,  the  whole  is  read 
11 137  and  439  thousandths  meters." 

NOTE.  The  pupil  will  get  the  clearest  notions  of 
decimals  by  taking  a  meter-stick  (which  is  divided 
in  tenths,  hundredths,  and  thousandths)  and  meas- 
uring given  lengths ;  such  as,  the  length  of  the  side 
of  the  room,  of  the  platform,  of  the  window-sill,  etc., 
etc.,  and  writing  down  the  result  in  each  case. 
Whenever  the  length  measured  is  less  than  a  meter, 
he  should  write  down  0,  and  after  it  the  decimal 
point,  then  the  actual  measure.  Thus,  if  the  length 
is  found  to  be  8  tenths  2  hundredths  and  7  thou- 
sandths, it  is  expressed  by  0.827,  and  read  "  eight  hundred  twenty 
seven  thousandths  of  a  meter." 


DECIMALS.  79 


74,  It  will  be  seen  that  1   tenth  =  10  hundredths,  1 
hundredth  =  10  thousandths ;    and,    conversely,  10  thou- 
sandths —  1  hundredth,  10  hundredths  —  1  tenth,  10  tenths 
=  1  unit ;  so  that  in  decimal  numbers,  as  in  integral  num- 
bers, 10  in  any  place  is  equal  to  1  in  the  next  place  to  the 
left,  and  1  in  any  place  is  equal  to  10  in  the  next  place  to 
the  right. 

Hence  figures  in  the  first  decimal  place  denote  tenths,  in 
the  second  place  hundredths,  in  the  third  place  thousandths, 
in  the  fourth  place  ten-thousandths,  in  the  fifth  place  hun- 
dred-thousandths, in  the  sixth  place  millionths,  and  so  on. 

75,  In  reading  decimals,  read  precisely  as  if  the  decimal 
were  an  integral  number,  and  add  the  name  of  the  lowest 
decimal  place.     It  is  best  to  pronounce  the  word  "  and  " 
at  the  decimal  point,  and  omit  it  in  all  other  places.    Thus, 
100.023  is  read  one  hundred  and  twenty-three  thousandths. 
Ambiguity  in  reading,  from  having  zeros  at  the  end  of  a 
decimal,  is  avoided  by  a  pause ;  thus,  0.300  is  read  three 
hundred  .  .  .  thousandths,  while  0.00003  is  read  three  .  .  . 
hundred-thousandths. 

76,  Read  the  following  numbers : 

0.3;  0.7;  0.65;  0.99;  37.5;  26.9;  425.312;  617.624; 
94.57  ;  83.28 ;  0.9 ;  0.96  ;  57.09  ;  3.207  ;  2.03  ;  3.045 ; 
40.7;  0.055;  0.074;  0.0215;  7.3945;  0.14875  ;  0.00005 ; 
2.000375;  100.015625;  3.7525;  2.1136257. 

77,  Express  in  the  decimal  notation  : 

Seven  tenths ;  nine  tenths ;  eleven  hundredths ;  eight 
hundredths ;  one  hundred  thirty-four  thousandths ; 
twenty-five  thousandths;  two  hundred  and  thirty-four 
thousandths ;  nineteen  and  forty-one  hundred-thou- 
sandths ;  twenty-five  and  sixteen  ten-thousandths ; 


80  DECIMALS. 


thirteen  and  two  hundred  one  hundred-thousandths ; 
six  hundred  fifty-eight  thousand  three  hundred  forty-two 
millionths  ;  eighty-six  and  eight  hundred  three  thousand 
three  hundred  four  millionths ;  three  and  twenty-nine 
hundredths ;  fifteen  and  six  hundred  seventy-one  thou- 
sandths ;  fifty-three  ten-thousandths ;  twenty-two  and 
sixty-seven  hundredths ;  fourteen  and  two  thousand 
three  hundred  fifty-one  ten-thousandths;  two  and  two 
hundred  nineteen  thousandths ;  three  and  one  hundred 
fifty-seven  thousandths. 

78,  Zeros  occurring  at  the  end  of  a  decimal  do  not  affect 
its  value.  Thus,  3.50700  means  3  units  +  5  tenths  +  0 
hundredths  +  7  thousandths  -f  0  ten-thousandths  +  0  hun- 
dred-thousandths, and  is,  therefore,  3  and  507  thousandths, 
the  same  as  3.507. 

78,    The  arrangement  and  method  of  working  employed 
in  decimals  is  precisely  like   that   employed   in   integral  s 
numbers,  the  decimal  point  being  the  only  new  considera- 
tion. 

ADDITION  OF  DECIMALS. 

Add  17.5163,  236.3,  1.7162,  0.00132, 
OPERATION. 

17.5163 
236.3 

1.7162 

0.00132 
255.53382 

Write  the  numbers  in  columns,  units  under  units,  tens 
under  tens,  tenths  under  tenths,  and  so  on,  so  that  the 
decimal  -points  will  fall  in  a  vertical  line,  and  add  as  in 
integral  numbers. 


DECIMALS.  81 


Ex.  53. 
Find  the  value  of: 

1.  2.514  +  3.7  +  9.6304  +  0.24876. 

2.  1.916  +  6.3  +  0.4782  +  9.35634. 

3.  0.415  +  8.0  +  6.3746  +  8.29426. 

4.  7.516  +  9.6  +  1.9238  +  7.21442. 

5.  7.03  +  7.2456  +  0.483  +  9.23579  +  8.3. 

6.  2.576  +  3.4203  +  1.5  +  6.27948  +  0.362357 

7.  3.29+15.671  +  0.0053  +  22.67. 

8.  14.2351  +  651  +  2.219  +  3.157. 

9.  213.7  +  2.913  +  14.769  +  0.007871. 

10.  1.4178  +  0.2  +  2.356709  +  1.14  +  2.0. 

11.  4.96  +  3.2728  +  0.7  +  3.54219  +  4.7. 

12.  1.198  +  3.5  +  7.635487  +  4.23  +  1.5724. 

13.  4.372  +  9.5  +  7.369248  +  1.72  +  3.2948. 

14.  0.4293  +  0.7  +  6.954326  +  3.14  +  7.005. 

15.  3.87  +  2.6493  +  0.8  +  2.63495  +  9.3. 

16.  6.9  +  5.71  +  0.0431  +  329.2  +  4.4. 

17.  3.571  +  0.008+12.51  +  649  +  3.051. 

18.  15.753  +  2.069+17.6143  +  3.2107. 

19.  1.1  +  20.02+13  +  2.845  +  1.0001. 

20.  31.826  +  3.471  +  0.004  +  45  +  0.6. 

21.  82.537  +  2000  +  1.354  +  0.006  +  13. 

22.  64.27  +  1.1  +  23  +  17.12  +  8.8. 

23 .  72.5  +  140  +  340.03  +  21.5715  +  4.00087. 

24.  0.96  +  7,3004  +  8010  +  0.00093  +  124650. 


82 


DECIMALS. 


SUBTRACTION  OF  DECIMALS. 

80,    Subtract  37.286  from  41.1325 ;  and  1.00523  from  9.3. 

OPERATION.  OPERATION. 

41.1325  9.30000 

37.286  1.00523 

3.8465  8.29477 

Write  the  subtrahend  under  the  minuend,  so  that  the 
decimal  points  may  fall  in  a  vertical  line.  If  the  number 
of  decimal  places  in  the  subtrahend  exceed  the  number  in 
the  minuend,  zeros  may  be  annexed  to  the  minuend,  as 
such  zeros  have  no  effect  on  its  value. 


1.  0.58-0.39. 

2.  0.67-0.59. 

3.  3.927-1.836. 

4.  4.825-1.763. 

5.  4.325-1.672. 

6.  6.283-3.576. 

7.  9.025-6.387. 

8.  6.275-3.829. 

9.  7.57-6.385. 

10.  9.26-2.375. 

11.  8.4-3.228. 

12.  9.5-2.732. 

13.  14.3846-4.8003. 

14.  3.4370-0.3045. 

15.  0.3290-0.0089. 

16.  136.0200-1.5423. 

17.  1.9990-0.063. 

18.  13.5298-10.0060. 


Ex.  54. 

19.  2.1808-0.0009. 

20.  1.9870-1.0873. 

21.  48.9370-30.3000. 

22.  0.9990-0.9009. 

23.  15.1409-3.8579. 

24.  5.9009-0.0909. 

25.  1.3993-0.9090. 

26.  10.1010-0.0999. 

27.  3.5-0.075. 

28.  517-0.0076. 

29.  1.325-0.4736. 

30.  192.3-17.294. 

31.  175.8-1.0024. 

32.  186.257-13.794. 

33.  0.715-0.70451. 

34.  1111.116-22.22222. 

35.  71.0047-9.0008167. 

36.  9161.0098-7149.16776. 


DECIMALS.  83 


MULTIPLICATION  OF  DECIMALS. 

81.  A  change  in  position  of  the  decimal  point  of  a  num- 
ber will  affect  the  local  value  of  each  figure  of  that  number. 
Thus,  if  in  place  of  79.213  we  write  792.13,  we  increase 
the  value  of  each   figure   ten-fold,   the  7  tens   become  7 
hundreds,  the  9  units  become  9  tens,  the  2  tenths  become 
2  units,  the  1  hundredth  becomes  1  tenth,  and  the  3  thou- 
sandths become  3  hundredths,  and,  as  the  value  of  every 
figure  is  increased  ten-fold,  the  entire  number  is  increased 
ten-fold.     If  the  decimal  point  is  moved  one  place  to  the 
left,  the  local  value  of  each  figure  is  diminished  ten-fold, 
and  consequently  the  value  of  the  entire  number  is  dimin- 
ished ten-fold.     Hence, 

To  multiply  a  decimal  by  10,  100,  1000,  etc.,  we  have 
only  to  move  the  decimal  point  in  the  multiplicand  as  many 
places  to  the  right,  annexing  zeros  if  necessary,  as  there 
are  zeros  in  the  multiplier. 

To  divide  a  decimal  by  10,  100,  1000,  etc.,  we  have  only 
to  move  the  decimal  point  in  the  dividend  as  many  places 
to  the  left,  prefixing  zeros  if  necessary,  as  there  are  zeros 
in  the  divisor. 

Thus,  100  X  36.123  =  3612.3,  and  1000  X  36.1  =  36100: 
36.123-^10  =  3.6123,  and  36.123 -*- 1000  =  0.036126. 

82,  To  multiply  a  number  by  0.1,  0.01,  0.001,  etc.,  we 
have,  by  the  definition    of  multiplication,   to    divide   the 
multiplicand  by  10,  100,  1000,  etc. ;  that  is,  to  remove  the 
decimal  point  one  place,  two  places,  etc.,  to  the  left. 

To  divide  by  0.1,  0.01,  0.001,  etc.,  we  have  only  to  move 
the  decimal  point  in  the  dividend  one  place,  two  places, 
etc.,  to  the  right. 

Thus,  0.1  X  86.32  =  8.632,  and  0.01  X  1.236  =  0.01236 ; 
86.32  -•-  0.1  =  863.2,  and  1.236  -*-  0.01  =  123.6. 


84  DECIMALS. 


83,    Multiply  123.826  by  3. 

Here  3x6  thousandths  =  18  thousandths,  or  1  hundredth  and  8 

OPERATION,    thousandths ;  the  8  therefore  is  written  in  the  thou- 

123  826      sandths'  column  ;  then,  3x2  hundredths  =  6  hundredths, 

3      which,  with  the  1  hundredth,  make  7  hundredths,  and 

S71  478      ^e  ^  *8  wr^ten  *n  the  bundredths'  column  ;  then,  3x8 

tenths  =  24  tenths,  or  2  units  and  4  tenths,  and  the  4 

is  written  in  the  tenths'  column  ;  then,  3x3  units  =  9  units,  which, 

with  the  2  units,  make  11  units,  and  so  on. 

Multiply  123.826  by  0.3. 

OPERATION. 

123.826 

03 

37.1478 

The  multiplier  0.3  =  3x0.1.  We  therefore  multiply  first  by  3, 
and  the  resulting  product  by  0.1.  But  multiplying  by  0.1  simply 
moves  the  decimal  point  in  the  product  one  place  to  the  left.  Hence, 
the  product  will  have  three  decimal  places  for  the  decimal  in  the 
multiplicand,  and  one  more  place  for  the  decimal  in  the  multiplier. 

Multiply  123.826  by  0.32. 

OPERATION. 

123.826 
0.32 
247652 
371478 


39.62432 

The  multiplier  0.32  =  32  X  0.01.  We  therefore  multiply  first  by 
32,  and  the  resulting  product  by  0.01.  But  multiplying  by  0.01 
simply  moves  the  decimal  point  in  the  product  two  places  to  the  left. 
Hence,  the  product  has  three  decimal  places  for  the  decimal  in  the 
multiplicand,  and  two  more  places  for  the  decimal  in  the  multiplier. 

In  the  multiplication  of  decimals,  therefore,  point  off  in 
the  product  as  many  decimal  places  as  there  are  in  the  mul- 
tiplicand and  multiplier  taken  together. 


DECIMALS. 


85 


Find  the  products  of : 

1.  5x0.3.  26. 

2.  8  X  0.27.  27. 

3.  12  X  0.375.  28. 

4.  15x0.256.  29. 

5.  9  X  0.7.  30. 

6.  6x0.75.  31. 

7.  16  X  0.284.  32. 

8.  11  X  0.386.  33. 

9.  10x0.65.  34. 

10.  100x0.721.  35. 

11.  1000x3.736.  36. 

12.  1000x0.074.  37. 

13.  10x0.99.  38. 

14.  100x0.615.  39. 

15.  1000x2.409.  40. 

16.  1000x0.055.  41. 

17.  0.5x37.  42. 

18.  0.9x99.  43. 

19.  0.25x428.  44. 

20.  0.36  X  7384.  45. 

21.  0.9x26.  46. 

22.  0.7  X  67.  47. 

23.  0.48  X  237.  48. 

24.  0.18x3692.  49. 

25.  0.312x425.  50. 


Ex.  55. 

0.716  x  388. 
0.725  x  96. 
0.085  x  88. 
0.624x617. 
0.358  X  776. 
0.145  X  48. 
0.017  X  44. 
57  X  9.4. 
26  X  3.8. 
3  x  972.3. 
65  x  87.2. 
2.8  x  83. 

3.2  x  64. 
7.8  x  369. 
3.7  X  815. 
1.44  x  9.6. 
2.88  X  4.8. 
3.21  x  72.5. 
2.16  X  40.7. 
3.26  x  4.37. 

2.03  x  3.207. 
2.472  X  9.525, 
3.264  x  3.045 
0.7  X  0.5. 

0.9  X  0.57. 


51.  0.45  x  0.57. 

52.  0.72x0.324 

53.  0.6  x  0.9. 

54.  0.8  x  0.96. 

55.  0.72  x  0.72. 

56.  0.36  x  0.648. 

57.  416  x  0.416. 

58.  57  X  0.015. 

59.  693  X  0.83. 

60.  4.625  X  7.14, 

61.  99.9  X  4.09. 

62.  753  x  0.672. 

63.  928  x  8.302. 

64.  56.704x0.413. 

65.  2.052x0.0037. 

66.  0.00948  X  29. 

67.  372  x  0.468. 

68.  9.43  x  0.054. 

69.  786  x  3.62. 

70.  0.632  X  85. 

71.  2.406  X  0.008. 

72.  6824  X  3.7. 

73.  42.53  X  0.685. 

74.  0.832  X  59. 

75.  763.24x4.078, 


86  DECIMALS. 


DIVISION  OF  DECIMALS. 

84,  In  Division,  if  the  dividend  and  divisor  are  both 
multiplied  or  both  divided  by  the  same  number,  the  quo- 
tient is  not  changed.  Thus,  18  -s-  6  =  3,  and  (when  both 
dividend  and  divisor  are  multiplied  by  2)  36  -s-  12  =  3. 
Again  (when  both  dividend  and  divisor  are  divided  by  2), 
9-^-3  =  3. 

If,  therefore,  the  divisor  contains  decimal  places,  we  may 
remove  the  decimal  point  from  the  divisor,  provided  we  carry 
the  decimal  point  in  the  dividend  as  many  places  to  the 
right  as  there  are  dermal  places  in  the  divisor. 

Divide  78.528  by  0.8. 

Here  the  decimal  point  is  removed  from  OPERATION. 
the  divisor,  and  the  decimal  point  in  the  divi-  8)  785.28 
dend  is  carried  one  place  to  the  right ;  that  is,  98.16 

both  dividend  and  divisor  are  multiplied  by  10. 

When  the  divisor  is  a  whole  number,  each  quotient 
figure  is  of  the  same  order  of  units  as  the  right-hand  figure 
of  the  partial  dividend  used  in  obtaining  it.  Hence,  the 
decimal  point  is  put  in  the  quotient  as  soon  as  the  decimal 
point  in  the  dividend  is  reached. 

Divide  28.3696  by  1.49. 

OPERATION. 

19.04 

149)2836.96 
149 
1346 
1341 


596 
596 

Here  the  decimal  point  is  removed  from  the  divisor,  and 
is  moved  two  places  to  the  right  in  the  dividend ;  in  other 
words,  both  dividend  and  divisor  are  multiplied  by  100. 


DECIMALS. 


87 


If  the  divisor  is  not  contained  in  the  dividend  without  a 
remainder,  ciphers  may  be  mentally  annexed  to  the  divi- 
dend, and  the  division  continued. 

Divide  0.39842  by  3.7164  to  four  decimal  places. 
OPERATION. 

0.1072 

37164)3984.2 
37164 
267800 
260148 


76520 

74328 

2192 

If  the  divisor  is  a  whole  number,  and  ends  in  zeros,  we 
may  cut  off  the  zeros  from  the  divisor,  and  move  the  deci- 
mal point  in  the  dividend  as  many  places  to  the  left  as  there 
are  zeros  cut  off. 

Divide  42.08  by  8000. 

OPERATION. 

8)0.04208 

0.00526 

Here  the  three  zeros  are  cut  off  from  the  divisor,  and  the 
decimal  point  in  the  dividend  is  moved  three  places  to  the 
left.  In  other  words,  both  divisor  and  dividend  are  divided 
by  1000. 

Ex.  66. 
Find  the  quotients  of : 

1.  34.24-^-4.28. 

2.  24.56^-6.14. 

3.  52.90-^-5.75. 


4.  37.576 -H 

5.  281.232- 


6.832. 
h  7.812. 


6.  97.524-^5.16. 

7.  738.0930-^-0.023. 

8.  5.18466 -- 1.02. 

9.  0.018  --9.6. 

10.  34.96818^0.381. 


88 

DECIMALS. 

11. 

0.003125  -f-  25. 

36. 

6  +  0.008. 

12. 

859.95  -i-  136.5. 

37. 

4.8  *•  0.00016. 

13. 

5.468  -*-  0.08. 

38. 

1562.5  H-  0.00025. 

14. 

0.04922-*-  0.0023. 

39. 

64  +  0.016. 

15. 

0.00044408  -f-  0.01  12. 

40. 

5.  76  -H  0.0048. 

16. 

0.20412^-0.0084. 

41. 

3.012-^-0.0006. 

17. 

0.07504  -f-  23.45. 

42. 

91844152.5  *  1.1575. 

18. 

0.00025  -*-  2.5. 

43. 

7  -5-  0.0035. 

19. 

0.03217  -  1250. 

44. 

0.39237  -H  0.319. 

20. 

171.99  -f-  27.3. 

45. 

0.3230864  H-  0.5072. 

21. 

0.012305-^-1.07. 

46. 

3.  I-H  0.0025. 

22. 

15.625  -f-  2.5. 

47. 

63.8406-^-0.18345. 

23. 

5.418-2.58. 

48. 

181.3  -f-  0.00037. 

24. 

0.59064  -«-  0.0276. 

49. 

12.5  -H  2.56. 

25. 

0.73807  -!-  0.023. 

50. 

284.7432^-0.00004. 

26. 

15.4546  -*-  0.019. 

51. 

130.4  -^  0.0004. 

27. 

6.7288  -H  64.7. 

52. 

1  13.4  -f-  0.0108. 

28. 

72.36  -*•  144. 

53. 

68.97516  -*-  0.9246. 

29. 

0.01124-*-  11.24. 

54. 

0.022185  •*-  0.0306. 

30. 

15.625  -*-  5. 

55. 

0.276766  -^  0.371. 

31. 

8.192-^0.00128. 

56. 

286-^-0.013. 

32. 

0.00512  -i-  2.048. 

57. 

0.10724-^-0.003125, 

33. 

0.00972  -H  0.0004. 

58. 

0.03  -*-  0.001. 

34. 

0.07504  -*-  23.45. 

59. 

105-^-43.75. 

35. 

15.21-11.7. 

60. 

8.468  -f-  0,0292. 

DECIMALS.  89 


Ex.  57. 

1.  To  enclose  a  certain  lot  225  yards  of  fence  are  needed. 

What  will  be  the  cost  of  the  fence  at  the  rate  of 
$0.50  a  yard? 

2.  A  section  of  land  costs  $49,878;  what  must  be  paid 

for  0.375  of  a  section  ? 

3.  When  0.7  of  a  ton  of  coal  is  worth  $6.30,  what  will  be 

the  cost  of  12.5  tons? 

4.  Coal  being  worth  $7.00  per  ton,  what  part  of  a  ton 

can  be  bought  for  $2.59? 

5.  If  a  man  can  build  0.45  of  a  rod  of  wall  in  one  hour, 

how  many  rods  will  4  men  build  in  3.8  days,  work- 
ing 7.5  hours  per  day  ? 

6.  Twelve  dozen  penknives  cost  .$90.     If  they  are  sold. 

at  $0.75  each,  what  will  be  the  gain  on  each? 

7.  Divide  $125. 15  by  $25.03. 

8.  Twelve  yards  of  velvet  cost  $150.      At  that   rate, 

what  must  be  paid  for  18  yards  ? 

9.  What  will  be  the  cost  of  9.75  cords  of  white  oak  wood 

at  the  rate  of  $  10  a  cord  ? 

10.  Twenty-five  hundred ths  of  a  farm  cost  $  5000 ;  what   V 

will  nine-tenths  of  it  cost  ? 

11.  A  merchant  bought  575  pounds  of  sugar  for  $51.75; 

he  sold  four-tenths  of  it  at  $0.11  per  pound,  and  the 
remainder  at  $0.125.     What  was  his  gain  ? 

12.  A  railroad  train  has  201  miles  to  run.     If  it  averages 

26.8    miles    per   hour,   how   many   hours   will  be 
required  ? 


90  DECIMALS. 


13.  Hiram  Jones   owes  to   one   man   $1000,  to   another 

$75.02,  to  another  $198.75,  and  to  a  fourth 
$325.50.  How  much  money  will  he  need  to  pay 
these  debts? 

14.  A  man  bought  80  yards  of  cotton  cloth  for  $18.40. 

Find  the  price  of  the  cloth  per  yard. 

15.  How  many  yards  of  calico  at  $0.125  per  yard  can  be 

bought  for  $45.625? 

16.  A  farmer  spent  $303.75  for  corn  at  $  1.35  a  bag.    How 

many  bags  of  corn  did  he  buy  ? 

17.  One  pound  of  dry  oak  wood  when  burnt  yields  0.022 

of  a  pound  of  ashes.  What  part  of  the  pound  dis- 
appears into  the  air  ? 

18.  Before  a  storm  the  mercury  in  a  barometer  fell  from 

30.292  inches  to  29.347  inches.  What  part  of  an 
inch  did  it  fall  ? 

19.  John  purchased  goods  to  the  amount  of  $7.48.     He 

gave  the  salesman  a  ten-dollar  bill.  How  much 
money  should  he  receive  ? 

20.  The  distance  from  New  York  to  Chicago  by  the  New 

York  Central  and  Lake  Shore  route  is  982.24  miles, 
and  the  distance  from  New  York  to  Buffalo  is 
441.75  miles.  What  is  the  distance  from  Buffalo 
to  Chicago  ? 

21.  Excursion  tickets  from  Boston  to  Fabyan's  and  return 

cost  $7.75,  from  Fabyan's  to  the  summit  of  Mt. 
Washington  and  return  $3.  If  a  person  takes 
this  trip  and  pays  $4.50  for  supper,  lodging,  and 
breakfast  at  the  Summit  House  on  Mt.  Washington, 
and  $1.25  for  dinner  at  Fabyan's  the  next  day,  how 
much  will  the  whole  trip  cost? 


DECIMALS.  91 


22.  From  an  ice-house  containing  1000  tons  of  ice  the 
owner  sold  at  different  times  242.765  tons,  92.325 
tons,  161.575  tons,  and  479.312  tons.  The  rest 
melted.  How  many  tons  melted  ? 

Ex.  58. 
To  find  the  cost  of  goods  sold  by  the  hundred, 

Point  off  two  places  for  decimals  at  the  right  of  the 
number  denoting  the  quantity,  and  multiply  the  price  of  a 
hundred  by  this  number. 

To  find  the  cost  of  goods  sold  by  the  thousand, 

Point  off  three  places  for  decimals  at  the  right  of  the 
number  denoting  the  quantity,  and  multiply  the  price  of  a 
thousand  by  this  number. 

NOTE.   M  is  often  used  for  thousand,  and  C  for  hundred. 

1.  What  will  9875  feet  of  boards  cost  at  $9  per  M  ? 

2.  At  $3.25  per  C,  what  must  be  paid  for  3784  feet  of 

Georgia  pine  ? 

3.  For  the  roof  of  a  building  8000  tiles  are  to  be  used. 

What  will  they  cost  at  $9.875  per  M  ? 

4.  Required  the  cost  of  98,762  laths  at  $0.35  per  C. 

5.  An  architect  estimates  that  1,500,784  bricks  will  be 

needed  for  a  schoolhouse.  What  will  they  cost  at 
$7.75  a  thousand? 

6.  What  must  be  paid  for  4879  paving-stones  at  $9.375 

a  hundred  ? 

7.  If  the  freight  from  New  York  to  Boston  is  $0.12  per 

hundred  pounds,  what  must  be  paid  on  five  boxes  of 
goods  weighing  respectively  348.25,  227.25,  429.25, 
396.125,  419.125  pounds? 


92  DECIMALS. 


8.  A  lumber  dealer  paid  $4.50  per  M  for  cedar  shingles 

and  sold  them  for  $5.75.     What  did  he  gain  on  35 
M? 

9.  A  farmer  contracted  for  boards  for  fencing  at  the  rate 

of  $12.375  per  M.     His  bill  for  lumber  amounted 
to  $61.875.     How  many  thousand  feet  did  he  buy? 

To  find  the  cost  of  goods  sold  by  the  ton, 
Point  off  from  the  right  of  the  number  denoting  the 
quantity  three  decimal  places,  multiply  the  price  of  a  ton 
by  this  number,  and  divide  this  result  by  2.  The  reason 
for  this  operation  follows  from  the  fact  that  two  thousand 
pounds  make  a  ton. 

Find  the  retail  price  of  7846  pounds  of  coal  at  $8.75  a 

ton. 

OPERATION. 

$8.75 
7.846 

5250 
3500 
7000 
6125 


2)68.65250 
$34.32625 

10.  What  must  be  paid  for  9785  pounds  of  plaster  at  $6.75 

per  ton  ? 

11.  If  25,000  pounds  of  plaster  cost  $131.25,  what  is  that 

per  ton  ? 

12.  A  dealer  in  New  York  retails  coal  at  $7.75  per  ton. 

If  a  ton  costs  $3.75  at  the  mine  and  $0.75   for 
freight,  what  will  he  make  on  8758  pounds  of  coal  ? 


DECIMALS.  -  93 


13.  At  $10.50  per  ton  what  is  the  cost  of  25,000  pounds 

of  plaster? 

14.  What  is  the  retail  price  of  coal  per  ton  when"  17,520 

pounds  are  sold  for  $  74.46  ? 

15.  How  many  pounds  of  plaster  at  $10.50  per  ton  can  be 

bought  for  $131.25? 

16.  An  errand  boy  receives  $2.75  a  week.     In  how  many 

weeks  will  he  earn  $44? 

17.  How  many  cords  of  pine  wood  at  $3.375  a  cord  must 

be  given  for  12  yards  of  broadcloth  at  $2.25   a 
yard? 

18.  The  milk  from  a  herd  of  75  cows  at  6  cents  a  quart 

amounted  in  one  summer  to  $2025.     How  many 
quarts  were  sold  ? 

19.  A  merchant  sold  3  pieces  of  matting,  each  containing 

45.5  yards,  at  $0.375  per  yard.     How  much  money 
did  he  receive  ? 

20.  If  a  man  earns  $12  a  week  and  spends  on  the  average 

$10.125,  in  how  many  weeks  will  he  save  $97.50? 

21.  A  grocer  bought  156  boxes  of  oranges  at  $5.625  each, 

and  sold  the  whole  for  $916.875.     How  much  did 
he  gain  ? 

22.  A  Western  farmer's  wheat  crop  at  $1.08  per  bushel 

brings  $831.60.     How  many  bushels  did  he  raise? 

23.  Find  the  cost  of  15,964  feet  of  boards  at  $39.25  a 

thousand. 

24.  Find  the  cost  of  19,500  laths  at  35  cents  a  hundred. 

25.  What  will  be  paid  for  shipping  1500  tons  of  wheat 

from  Buffalo  to  New  York  at  the  rate  of  5  cents  a 
bushel  ?     (A  bushel  of  wheat  weighs  60  pounds.) 


94  DECIMALS. 


Ex.  59. 

1.  Find  the  price  of  30  Parian  statuettes  at  $8.875  each. 

2.  In  February,  1884,  the  number  of  days  during  which 

rain  fell  in  New  England  was  22,  and  the  amount 
which  fell  was  4.57  inches.  Find  the  daily  average 
for  the  22  days. 

3.  How  many  acres  are  in   a  park  containing   0.08  of 

115.1875  acres? 

4.  If  31.75  rods  of  fence  are  made  for  $10.90,  what  is 

the  cost  of  a  rod  ? 

5.  On  a  certain  day  in  February,  1884,  the  thermometer 

at  the  highest  was  51.1°,  and  at  the  lowest  29.4°. 
Find  the  difference. 

6.  Of  100  parts  of  matter  in  beans,  sugar  and  gum  form 

61.10,  other  vegetable  matter  forms  31.55,  and 
moisture  5  parts.  Of  how  many  parts  does  the 
remainder,  which  is  mineral  matter,  consist? 

7.  If  0.1571  of  the  weight  of  superphosphate  is  organic 

matter,  find  the  weight  in  tons  of  organic  matter  in 
80  tons  of  superphosphate. 

8.  In  January,  1884,  the  barometer  at  the  highest  was 

30.543  inches,  and  at  the  lowest  28.843  inches. 
Find  the  difference. 

9.  A  cubic  inch  of  pure  water  weighs  252.458  grains. 

Find  the  weight  in  grains  of  a  cylindrical  inch, 
which  is  0.7854  of  a  cubic  inch. 

10.  Divide  $31.40  among  6  men  and  11  youths,  giving  a 
youth  0.525  of  a  man's  share.  What  is  each  man's 
share  ? 


DECIMALS.  95 


11.  .Four  men  together  paid  $20,000  for  some  land.     The 

first  puts  in  $2350,  the  second  $5820.35,  the  third 
$7640.75.  How  much  must  the  fourth  man  pay  ? 

12.  What  will  be  the  cost  of  uniforms  for  a  base-ball  nine 

at  $2.87  for  each  uniform  ? 

13.  At  $15.87  a  ton,  what  will  be  the  value  of  637  tons 

of  hay  ? 

14.  If  peaches  are  worth  $1.25  a  basket,  and  it  takes  3 

dozen  for  a  basket,  what  is  the  value  of  2892  dozen 
peaches  ? 

15.  If  964  baskets  of  peaches  are  sold  for  $1301.40,  what 

is  the  price  per  basket  ? 

16.  If  324  men  contribute  together  $2647.08,  what  is  the 

contribution  of  each  ? 

17.  A  boy  picks  blueberries  in  a  pasture,  giving  to  the 

owner  of  the  pasture  for  the  privilege  1  quart  out 
of  every  8  quarts.  In  2  days  he  picks  48  quarts, 
and  sells  his  share  of  the  berries  for  $3.78.  What 
did  he  get  a  quart? 

18.  If  150  men  work  on  a  railroad  at  the  same  price  per 

day,  and  if,  at  the  end  of  the  week,  they  all  together 
receive  $1575,  what  price  per  day  does  each  man 
receive? 

19.  If  a  kite-string  is  213.86  feet  long,  and  the  kite  breaks 

away  and  carries  off  94.38  feet  of  the  string,  how 
much  will  be  left  ?  How  much  more  must  be 
bought  to  make  up  1000  feet? 

20.  At  $9.17  a  barrel,  how  many  barrels  of  flour  can  be 

bought  for  $876.35,  and  how  much  money  will  be 
leftover? 


96  DECIMALS. 


Ex.  60. 

If  the  length  of  the  diameter  of  a  circle  is  multiplied  by 
3.1416,  the  product  is  the  length  of  the  circumference. 
1.    Find  the  length  in  inches  of  the  circumference  of  a 
circle  if  the  diameter  is  6  inches. 


2.  Find  the  length   in  inches  of  the  circumference  of  a 

circle  if  the  diameter  is  17  inches. 

3.  If  a  carriage  wheel  is  4  feet  in  diameter,  what  is  its 

circumference  in  feet? 

4.  If  the  carriage  wheel  in  Example  3  rolls  on  the  ground 

without  slipping,  how  many  feet  will  it  go  in  turn- 
ing 27  times? 

5.  How  many  times  will  the  carriage  wheel  in  Example  3 

turn  in  going  1160  feet? 

If  the  length  of  the  circumference  of  a  circle  is  divided  by 
3.1416,  the  quotient  is  the  length  of  the  diameter. 

6.  Find  the  diameter  in  feet  of  a  circle  if  its  circumfer- 

ence is  1000  feet. 

7.  Find  the  diameter  in  feet  of  a  wheel  which  revolves 

19.5  times  in  going  253.5  feet. 

8.  If  the  circumference  of  a  circle  is  198  yards,  what  is 

its  diameter  in  yards  ? 


DECIMALS.  97 


9.    How  deep  is  a  well  if  the  wheel  whose  diameter  is  2 
feet  makes  10  revolutions  in  raising  the  bucket? 

10.  If  a  carriage  wheel  makes  440  revolutions  in  travelling 

a  mile  (5280  feet),  what  is  its  diameter  in  feet  ? 

11.  If  1964.52  bushels  of  corn  are  to  be  put  into  bags  hold- 

ing 2.14  bushels  each,  how  many  bags  will  it  take? 

12.  A  boy  has  3  pieces  of  twine:  one  is  58.74  feet  long, 

another  is  97.86  feet,  and  a  third  57.26  feet.  How 
long  a  kite-string  can  he  make,  making  no  allow- 
ance for  knots  ? 

13.  Boys   in   playing  hare  and  hound  run  3.876  miles. 

The  hares  drop  a  piece  of  paper  every  4.75  feet  on 
the  average.  How  many  pieces  do  they  drop  ?  A 
mile  is  5280  feet. 

14.  If  a  man  earns  $23.25  a  day,  how  many  days  will  it 

take  him  to  earn  $1964.87? 

15.  A  grain  merchant  bought  corn  at  60  cents  and  rye  at 

75  a  bushel.  He  bought  the  same  number  of  bushels 
of  both  kinds  of  grain  and  paid  for  both  together 
$607.50.  How  many  bushels  of  each  kind  did  he 
buy? 

16.  When  potatoes  are  worth  $0.77  per  bushel,  and  corn 

$1.10,  how  many  bushels  of  corn  should  a  farmer 
receive  in  exchange  for  50  bushels  of  potatoes  ? 

17.  How  many  gallons  of  231  cubic  inches  are  contained 

in  a  cubic  foot  (1728  cubic  inches)  ? 

18.  How  many  cubic  feet  are   contained  in  a  bushel,  a 

bushel  containing  2150.42  cubic  inches? 

19.  For  $7624.13  how  many  tons  of  hay  can  be  bought  at 

$18.75  a  ton? 

20.  The  large  wheel  of  a  bicycle  is  14.37  feet   around. 

How  many  times  will  it  turn  in  going  a  mile  (5280 
feet)  ? 


CHAPTER  VII. 

MULTIPLES  AND   MEASURES. 

85,  When  the  multiplier  is  an   integral   number,  the 
product  is  called  a  multiple  of  the  multiplicand ;  and,  in 
division,  when   the  quotient  is  an   integral    number,  the 
divisor  is  called  a  measure  of  the  dividend.   Thus,  8  X  7  =  56 ; 
the  number  56  is  a  multiple  of  7.     Again,  56  -f-  7  =  8 ;  the 
number  7  is  a  measure  of  56. 

86,  A  number  which  cannot  be  divided  by  any  other 
number  except  unity  without  remainder  is  called  a  prime 
number, 

Thus,  2,  3,  5,  7,  11,  13,  17,  19,  23,  29,  31,  etc.,  are  prime 
numbers. 

87,  Other  numbers  are  each  the  product  of  a  fixed  set  of 
prime  numbers,  and  are  called  composite  numbers, 

88,  Numbers  which  can  be  divided  by  2  without  re- 
mainder are  called  even  numbers ;  and  all  other  numbers 
are  called  odd  numbers,     Even  numbers  end  in  2,  4,  6,  8, 
or  0;  odd  numbers  end  in  1,  3,  5,  7,  or  9. 

89,  By  way  of  distinction,  when  a  number  is  used  with- 
out reference  to  any  designated  unit,  it  is  called  an  abstract 
number ;  and,  when  used  with  reference  to  a  specified  unit, 
it  is  called  a  concrete  number, 

Thus,  5,  7,  8  are  abstract  numbers,  and  5  horses,  7  chairs, 
8  dollars  are  called,  by  way  of  distinction,  concrete  numbers. 


MULTIPLES   AND   MEASURES.  99 

90,     To  factor  a  composite  number  is  to  separate  the 
number  into  its  factors. 

Find  the  prime  factors  of  144. 
2144 


72 
36 


9 


3 

That  is,  144  -  2  x  2  x  2  x  2  x  3  x  3. 

91.  To  avoid  the  necessity  of  writing  long  rows  of  equal 
factors,  a  small  figure  called  the  exponent  is  written  at  the 
right  of  a  number  to  show  how  many  times  the  number  is 
taken  as  a  factor. 

Thus,  2x2x2x2x3x3  is  written  24  X  32. 
The  expression  2*  is  called  the  fourth  power  of  2,  and  32 
is  called  the  second  power  of  3. 

92.  It  is  evident  from  §  90  that  the  method  of  separating 
a  composite  number  into  its  prime  factors  is, 

Divide  the  given  number  by  any  prime  number  that  is 
contained  in  it  without  remainder  ;  then  the  quotient  by  any 
prime  number  that  is  contained  in  it  without  remainder ;  and 
so  on  until  the  quotient  is  itself  a  prime  number.  The 
several  divisors  and  the  last  quotient  are  the  prime  factors. 

If  no  prime  factor  is  found  before  the  quotient  becomes 
equal  to  or  less  than  the  divisor,  the  number  is  prime. 

03,  The  following  tests  are  useful  for  determining  with- 
out actual  division  if  a  number  contains  certain  factors  : 

1.  A  number  is  divisible  by  2  if  its  last,  or  right  hand, 
digit  is  even. 

2.  A  number  is  divisible  by  4  (22)  if  the  number  denoted 
by  the  last  two  digits  is  divisible  by  4. 


100  MULTIPLES   AND   MEASURES. 

3.  A  number  is  divisible  by  8  (23)  if  the  number  denoted 
by  the  last  three  digits  is  divisible  by  8. 

4.  A  number  is  divisible  by  3  if  the  sum  of  its  digits  is 
divisible  by  3. 

5.  A  number  is  divisible  by  9  (32)  if  the  sum  of  its  digits 
is  divisible  by  9. 

6.  A  number  is  divisible  by  5  if  its  last  digit  is  either 
5  orO. 

7.  A  number  is  divisible  by  25  (52)  if  the  number  de- 
noted by  the  last  two  digits  is  divisible  by  25. 

8.  A  number  is  divisible  by  125  (53)  if  the  number  de- 
noted by  the  last  three  digits  is  divisible  by  125. 

9.  A  number  is  divisible  by  6  if  its  last  digit  is  even, 
and  the  sum  of  its  digits  is  divisible  by  3. 

10.  A  number  is  divisible  by  11  if  the  difference  between 
the  sum  of  the  digits  in  the  even  places  and  the  sum  of  the 
digits  in  the  odd  places  is  either  0  or  a  multiple  of  11. 

Ex.  61. 

Find  the  prime  factors  of: 

1.  32;  48:  56;  60;  75;  63;  92;  44;  88;  72;  84;  85. 

2.  51;  69;  68;  87;  54;  98;  74;  90;  86;  70;  42;  62. 

3.  112;  140;  132;  216;  162;  176;  252;  240;  360;  384. 

4.  484;  476;  512;  525;  560;  572;  632;  648;  696;  720. 

5.  748;  775;  824;  876;  888;  948;  960;  925;  117;  119. 

94,    The  number  1.56  may  be  put  in  the  form  of  156  X  .01, 
and  thus  separated  into  22  X  3  X  13  X  .01. 

Ex.  62. 

Find  the  prime  factors  of: 

'l.    1.05;  12.5;  14.3;  1.65;  19.2;  2.42;  62.4;  27.5. 
2.    34.3;  5.39;  62.1;  118.8;  1.331;  1.452;  1.584;  92.4. 


MULTIPLES   AND   MEASURES.  101 


GREATEST  COMMON  MEASURE. 

95.  The  measures  of  12  are  1,  2,  3,  4,  6,  12,  and  the 
measures  of  18  are  1,  2,  3,  6,  9,  18.     These  two  numbers 
have  the  measures  1,   2,   3,   G  in  common,  and  of  these 
measures  6  is  the  greatest. 

The  measures  that  two  or  more  numbers  have  in  common 
are  called  their  common  measures,  and  the  greatest  of  these 
is  called  their  Greatest  Common  Measure,  which,  for  the  sake 
of  brevity,  is  denoted  by  the  letters  G.  C.  M. 

If  two  or  more  numbers  have  no  common  measure  they 
are  said  to  be  prime  to  each  other.  Thus,  27  and  125  are 
prime  to  each  other . 

96,  The  prime  factors  of  12  are  22,  3. 
The  prime  factors  of  18  are  2,  32. 

The  prime  factors  common  to  12  and  18  are  2,  3.  The 
G.C.M.  of  12  and  18,  namely  6,  is  2  X  3. 

That  is,  the  G.  C.  M.  of  two  or  more  numbers  is, 

The  product  of  the  prime  factors  common  to  the  numbers, 
each  prime  factor  having  the  least  exponent  that  it  has  in 
any  one  of  the  numbers. 

Hence,  to  find  the  G.C.M.  of  two  or  more  numbers, 

Separate  the  numbers  into  their  prime  factors. 

Select  the  lowest  power  of  each  factor  that  is  common  to 
the  given  numbers,  and  find  the  product  of  these  powers. 

Find  the  G.C.M.  of  84,  105,  63. 


84 
42 
21 


105 
35 


7 

7  v 

84  =  22  X      <  7x       105  =  \  x  5  x  7.        63  =     x  7. 
Hence,  G.C.M.  =  3  X  7  or  21> 


01  \  v  \ 


102  MULTIPLES   AND   MEASURES. 

97,    Common  factors  of  two  or  more   numbers  may  be 
taken  out  of  the  numbers  simultaneously,  as  follows  : 


84 

105 

63 

28 

35 

21 

4 

5 

3 

The  number  3  is  seen  to  be  a  factor  of  all  the  numbers,  and  7  of 
the  resulting  quotients  28,  35,  21.  The  quotients  4,  5,  and  3  have  no 
common  factor.  Therefore,  3  and  7  are  the  only  common  factors,  and 
the  G.C.M.  is  3  x  7,  or  21. 

Ex.  63. 
Find  the  G.C.M.  of: 

1.  48,  128.  15.  216,  360.  28.  336,  884. 

2.  36,  90.  16.  279,  403.  29.  352,  364. 

3.  64,  256.  17.  294,  378.  30.  1344,  1536. 

4.  24,  105.  18.  210,  294.  31.  21,  35,  56. 

5.  125,  600.  19.  182,  196.  32.  42,  133,  56. 

6.  56,  138.  20.  225,  375.  33.  32,  48,  128. 

7.  63,  108.  21.  195,  299.  34.  27,  36,  108. 

8.  40,  600.  22.  288,  360.  35.  96,  48,  60, 108, 

9.  65,  91.  23.   133,  152.  36.  33,  297,  198. 

10.  39,  273.  24.  23,  111.  37.  56,  63,  315. 

11.  56,  126.  25.  352,  384.  38.  75,  225,  500. 

12.  232,  493.  26.  123,  579.  39.  232,  290,  493. 

13.  365,  511.  27.  960,  1536.  40.  365,  511,  803. 

14.  148,  592. 

98,  When  it  is  required  to  find  the  Gr.C.M.  of  two  or 
more  numbers  that  cannot  readily  be  separated  into  factors, 
the  method  to  be  employed  is  as  follows : 


MULTIPLES   AND   MEASURES.  103 

Find  the  G.C.M.  of  63  and  217. 

OPERATION. 

63)217(3 
189 

28)63(2 
56 

7)28(4 
28 

Therefore,  the  G.C.M.  is  7. 
Hence,  by  this  method, 

Divide  the  greater  number  by  the  less,  and  then  the  divisor 
by  the  remainder  left,  and  so  on  till  there  is  no  remainder. 
The  last  divisor  will  be  the  G.  C.  M.  required. 

To  find  the  G.C.M.  of  several  numbers,  find  the  G.C.M. 
of  two  of  the  numbers,  then  of  that  result  and  a  third  num- 
ber, and  so  on.  The  last  G.C.M.  is  the  one  required. 

Ex.  64. 
Find  the  G.C.M.  of: 

1.  342,  665.     6.  1131,  2639.  11.  3927,  5049. 

2.  841,  899.     7.  9889,  986.  12.  1287,  1551. 

3.  961,  1178.    8.  1792,  1832.  13.  1537,  1802. 

4.  1243,  1469.   9.  1847,  1792.  14.  3056,  3629. 

5.  1001,  1287.   10.  1850,  1517.  15.  2108,  3813. 

16.  4844,  5536.  22.  216,  105,  405. 

17.  696,  1305.  23.  112,  192,  128. 

18.  232,  3219.  24.  168,  132,  352. 

19.  949,  1387.  25.  198,  495,  209,  660. 

20.  1081,  1311.  86.  146,  730,  365,  219. 

21.  4067,  2573,  27.  924,  378,  612,  246. 


104  MULTIPLES   AND   MEASURES. 


LEAST  COMMON  MULTIPLE. 

99,  The  multiples  of  3  are  3,  6,  9,  12,  15,  18,  21,  24,  27, 
30,  etc. 

The  multiples  of  5  are  5,  10,  15,  20,  25,  30,  35,  etc. 
The  multiples  common  to  3  and  5  are  15,  30,  etc.,  and  of 
these  15  is  the  least. 

100,  The  multiples  that  two  or  more  numbers  have  in 
common  are  called  their  common  multiples,  and  the  least  of 
these  is  called  their  Least  Common  Multiple,  which  is  denoted 
by  the  letters  L.C.M. 

Find  the  L.C.M.  of  7,  8,  9,  21. 

The  L.C.M.  of  7,  8,  9,  21,  must  contain  the  factor  7,  or 
it  would  not  be  a  multiple  of  7.  It  must  also  contain  23  to 
be  a  multiple  of  8,  and  32  to  be  a  multiple  of  9.  It  must 
contain  the  factors  3  and  7  to  be  a  multiple  of  21.  That  is, 
the  L.C.M.  of  7,  8,  9,  21,  is  the  product  of  the  factors  7,  2s, 
32  ;  therefore,  it  is  7  X  8  X  9  =  504.  Hence, 

To  find  the  L.  C.  M.  of  two  or  more  numbers, 
Separate  each  number  into  its  prime  factors. 
Select  from  these  the  highest  power  of  each  factor,  and  find 
the  product  of  these  powers. 

Find  the  L.C.M.  of  16,  21,  24,  30,  32. 
16  =  24, 
21  =  3  x  7, 


30  =  2  x  3  x  5, 
32  =  25. 

Hence,  the  L,  C.  M,  =  25  x  3  x  5  X  7  «  3360, 


MULTIPLES   AND   MEASURES.  105 

The  L.C.M.  of  16,  21,  24,  30,  32,  may  be  found  as  fol- 
lows : 


21     24     30    32 


21     12     15     16 


21       6     15      8 


21       g     15      4 


7  54 

Hence,  the  L.C.M.  =  23  x  3  x  7  X  5  x  4  •=  3360. 

Since  16  is  a  measure  of  32,  it  is  elided,  for  any  multiple  of  32  is 
also  a  multiple  of  16.  The  even  numbers  are  divided  by  2  ;  the  quo- 
tients and  the  odd  numbers  are  written  below  the  horizontal  line.  This 
operation  is  repeated  so  long  as  2  is  a  measure  of  more  than  one 
number.  In  the  fourth  line  3,  a  measure  of  15,  is  elided.  The  divi- 
sion by  3  leaves  in  the  fifth  line  the  numbers  7,  5,  4,  which  are  prime 
to  each  other. 

Therefore,  the  factors  contained  in  the  numbers  are  2,  2,  2,  3,  and 
7,  5,  4. 

Hence,  the  L.C.M.  =  2x2x2x3x7x5x4  =  3360. 

When  two  or  more  numbers  are  prime  to  each  other,  their 
L.C.M.  is  their  product.  Thus,  the  L.C.M.  of  3,  5,  7,  is 
3x5x7. 

Ex.  65. 
Find  the  L.C.M.  of: 

1.  3,  9,  27,  54.  9.  22,  44,  88,  108. 

2.  6,  9,  24,  40.  10.  15,  30,  45,  60. 

3.  144,  12,  18,  96.  11.  8,  16,  24,  32. 

4.  3,  8,  12,  22.  12.  13,  15,  26,  39. 

5.  16,  30,  48,  15.  /""13.  7,  17,  51,  119. 

6.  12,  24,  63,  84.  .  - 14.  8,  6,  28,  32. 

7.  9,  27,  33,  54.  15.  4,  21,  42,  63. 

8.  12,  20,  36,  54,  16.  3,  6,  18,  22, 


106  MULTIPLES   AND   MEASURES. 

17.  5,  15,  24,  30.  27.  16,  24,  13,  7. 

18.  7,  2,  3,  5.  28.  5,  9,  14,  96,  128. 

19.  13,  5,  2,  26.  29.  32,  36,  49,  56,  42. 

20.  5,  10,  20,  100.  30.  20,  24,  25,  27,  45. 

21.  19,  38,  2,  76.  31.  28,  30,  32,  36,  42. 

22.  3,  9,  27,  81.  32.  35,  40,  42,  49,  28. 

23.  6,  18,  22,  99.  33.  14,  18,  21,  32,  28. 

24.  18,  26,  117,  312.  34.  24,  27,  32,  36,  56. 

25.  13,  26,  39,  65.  35.  21,  24,  27,  28,  35. 

26.  9,  36,  3,  45.  36.  28,  32,  56,  72,  96. 

101,  If  the  given  numbers  are  large,  and  contain  no 
prime  factors  that  can  readily  be  detected,  the  common 
factors  may  be  obtained  by  the  process  of  finding  the 
G.G.M.  under  like  circumstances. 

Find  the  L.C.M.  of  1189  and  2117. 

1189)2117(1  ) 

1189 

928)1189(1 
928 

261)928(3 
783 

145)261(1 
145 

116)145(1 
116 
29)116(4 

116 

Hence,  the  G.C.M.=29. 
Therefore,  1189  =  29  X  41 ;  2117  =  29  X  73. 
Therefore,  the  L.C.M.  =  29  X  41  X  73  =  1189  X  73. 


MULTIPLES  AND   MEASURES.  107 

From  this  process  it  will  be  seen  that : 

The  L.C.M.  of  two  numbers  may  be  found  by  dividing 
one  of  the  numbers  by  their  G.C.M.,  and  multiplying  the 
quotient  by  the  other  number. 

Ex.  66. 

Find  the  L.C.M.  of: 

1.  510  and  595.  7.  187  and  255. 

2.  217  and  643.  8.  1261  and  663. 

3.  506  and  308.  9.  255  and  357. 

4.  296  and  407.  10.  432  and  840. 

5.  645  and  275.  11.  949  and  2920. 

6.  468  and  923.  12.  1247  and  1769. 

Ex.  67. 

1.  A  farmer  owns  132  acres  of  wood-land,  and  99  acres 

of  pasture  ;  he  wishes  to  divide  them  into  equal  lots 
of  the  largest  possible  size.  How  many  lots  will 
there  be,  and  what  number  of  acres  in  each  one  ? 

2.  A  merchant  has  75  yards  of  one  kind  of  silk,  225  of  a 

second,  and  200  of  a  third  ;  if  he  cut  them  into  dress 
patterns  of  equal  size,  what  is  the  largest  number 
of  yards  which  each  pattern  can  contain  ? 

3.  Simeon  Jones  has  260  bushels  of  rye,  384  of  oats,  and 

416  of  wheat.  He  sends  his  grain  to  market  in  bags 
of  equal  size.  What  is  the  greatest  number  of 
bushels  which  each  bag  can  hold,  provided  there  is 
no  mixture  of  the  different  kinds  of  grain  ? 

4.  What  width  of  carpet  will  fit  three  rooms,  the  first  15 

feet  wide,  the  second  21  feet,  and  the  third  33  feet  ? 


108  MULTIPLES   AND   MEASURES. 

5.  A  milk-man  has  four  different  measures,  holding  2,  3, 

5,  and  6  quarts  respectively.  What  is  the  smallest 
vessel  that  can  be  exactly  filled  by  each  of  them  ? 

6.  Find  the  length  of  the  greatest  line  that  exactly  meas- 

ures the  sides  of  an  enclosure  216  yards  long  and 
111  broad. 

7.  Find  the  contents  of  the  smallest  vessel  that  may  be 

filled  by  using  a  4-quart,  a  5-quart,  or  a  6-quart 
measure. 

8.  Two  apprentices  carry  1147  and  961  ivory  balls,  re- 

spectively, from  the  workshop  to  the  showroom. 
The  balls  are  carried  in  baskets  of  equal  size,  which 
are  filled  and  emptied  several  times.  How  many 
balls  in  a  basketful  ? 

9.  Find  the  shortest  distance  that  three  lines,  8  feet,  9 

feet,  and  12  feet  long  will  exactly  measure. 

CANCELLATION. 

102.    Cancellation  is  the  operation  of  striking  out  equal 
factors  from  the  dividend  and  the  divisor. 

12  X  3 

Find  the  quotient  of  — — - 

o  X  ^ 


The  3  in  the  divisor  cancels  the  3  in  the  dividend.  Then  the  fac- 
tor 2  of  the  divisor  is  cancelled,  and  2  is  cancelled  from  12  in  the 
dividend.  The  resulting  dividend  is  6x1,  and  the  divisor  1x1, 
and  therefore  the  quotient  is  6. 

Cancellation  is  used  to  shorten  arithmetical  work, 


MULTIPLES   AND   MEASURES.  109 


Ex. 
l  9  X  18  X  24 

68. 
11. 

12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 

39  X  18  X  9 

12  X  3  X  6 
0  15  X  36  X  27 

13  X  27  X  2 
50  X  9  X  84 

9x12x5 
12  X  6  X  25 

12  X  3  X  75 

144  X  81  x  2 

5  X  12  X  3 
d  19  X  27  X  30 

27x96 
625x9 

9x15x3 
45  x  16  X  9 

75x3 
75  x  9  X  96 

8x15x3 
32  x  49  X  6 

12  x  15  x  9 
87  x  15  x  9 

3  x  16  X  7 
7     1728 

5  x  9  x  29 
84  X  91  x  8  x  20 

'  12x12x12 
8  25  x  6  X  28 

32  X  60  x  13 
169  X  196  X  16 

6  X  4  X  35 
9  6  X  54  x  7  X  24 

26  X  14  X  56 
60  X  9  X  90  X  7 

7  X  8  X  9  X  12 
10   64  x  105  x  12 

42  x  27  X  15 
13  X  19  X  17  X  20 

21  X  12  X  8  X  8 

17  X  260  X  4  X  361 

PRACTICAL  APPLICATIONS. 

A  man  carried  to  a  store  49  bushels  of  potatoes,  which 
he  sold  at  35  cents  a  bushel,  and  took  his,pay  in  sugar  at  7 
cents  a  pound.    How  many  pounds  of  sugar  did  he  receive  ? 
SOLUTION.   1  bushel  of  potatoes  is  sold  for  35  cents. 
49  bushels  are  sold  for  49  X  35  cents. 
1  pound  of  sugar  is  bought  for  7  cents. 
Number  of  pounds  bought  for  49x35  cents  is49*35- 


04* 

Ans.  245  pounds, 


110  MULTIPLES   AND   MEASURES. 


Ex.  69. 

1.*  How  many  yards  of  cloth,  at  $3  a  yard,  can  be  bought 
for  12  tons  of  hay,  at  $  15  per  ton. 

2.  How  many  pairs  of  boots,  at  $4  a  pair,  can  be  bought 

for  40  pounds  of  butter,  at  40  cents  per  pound  ? 

3.  How  many  jars  of  lard  of  36  pounds  each,  at  8  cents 

per  pound,  must  be  given  for  16  pieces  of  cloth  con- 
taining 24  yards  each,  at  48  cents  a  yard? 

4.  How  many  coats,  at  $4  each,  can  be  bought  for  32 

yards  of  broadcloth,  at  $2.50  a  yard? 

5.  A  milkman  having  30  cows  which  daily  give  8  quarts 

each,  sells  the  milk  at  5  cents  per  quart.  How  many 
pieces  of  cloth  containing  40  yards  each,  at  12  cents 
per  yard,  ought  he  to  receive  for  the  milk  of  6  days? 

6.  A  market  gardener  sold  16  lots  of  celery,  120  bunches 

in  each,  at  28  cents  per  bunch ;  how  many  240-pound 
barrels  of  sugar,  at  8  cents  a  pound,  will  the  celery 
pay  for? 

7.  John  Peters  sold  9  firkins  of  butter  weighing  78  pounds 

each,  at  25  cents  per  pound ;  how  many  pieces  of 
matting  having  45  yards  in  a  piece,  at  30  cents  per 
yard,  should  he  receive? 

8.  A  workman  has  received  for  15  days'  work  of  7  hours 

each,  21  dollars.  How  much  would  he  receive  for 
19  days'  work  of  5  hours  each  ? 

9.  Thirty  workmen  have  made  in  9  days  215  yards  of 

wall.  At  the  same  rate,  how  much  would  36  work- 
men make  in  15  days? 

10.  A  telegraph  operator  transmits  50  words,  averaging  4 
letters  each,  in  the  space  of  5  minutes.  At  the  same 
rate,  how  many  minutes  will  be  required  to  send  a 
dispatch  of  120  words,  averaging  5  letters  each  ? 


CHAPTER  VIII. 

COMMON    FRACTIONS. 

103,     What  is  the  name  of  one  of  the  parts  when  a  unit 
is  divided  into : 

1.  Two  equal  parts?  6.  Eight  equal  parts? 

2.  Three  equal  parts?  7.  Ten  equal  parts? 

3.  Four  equal  parts?  8.  Twelve  equal  parts ? 

4.  Five  equal  parts?  9.  Sixteen  equal  parts? 

5.  Six  equal  parts?  10.  Twenty  equal  parts? 

A  unit  contains  how  many : 

1.  Halves?              5.    Sixths?  9.  Sevenths? 

2.  Thirds?               6.    Eighths?  10.  Ninths? 

3.  Fourths?             7.    Tenths?  11.  Elevenths? 

4.  Fifths?                8.    Twelfths?  12.  Thirteenths? 

13.  Twentieths?  15.    Thirtieths? 

14.  Twenty-fourths?  16.    Thirty-seconds? 

When  a  unit  is  divided  into  twelve  equal  parts,  what  is 
the  name  of: 

1.  One  part?  4.    Two  parts?         7.    Eight  parts? 

2.  Three  parts?      5.    Four  parts?        8.    Nine  parts? 

3.  Five  parts?         6.    Six  parts?  9.    Twelve  parts? 


112 


COMMON   FRACTIONS. 


Express  in  figures 

1.  Three-sevenths. 

2.  Five-ninths. 

3.  Seven-eighths. 

4.  Five-twelfths. 
Read:  ft,  ft,  A 


,  A, 


5.  Seven-sixteenths. 

6.  Five-eighteenths. 

7.  Four-elevenths. 

8.  Nine-twentieths. 

H-  tt-  ii  A-  A- 


104,  The  expression  •£  means  : 

I.  Seven  of  the  parts  when  a  unit  has  been  divided  into 
nine  equal  parts. 

II.  One-ninth  of  seven  units  ;  for,  if  seven  units  be  divided 
into  nine  equal  parts,  one  of  these  parts  will  be  seven  times 
as  great  as  one  of  the  parts  obtained  by  dividing  one  unit 
into  nine  equal  parts. 

III.  The  quotient  of  seven  divided  by  nine. 

105,  In  the  fraction  •£,  the  lower  figure  shows  the  num- 
ber of  equal  parts  into  which  the  whole  has  been  divided, 
and  is  therefore  a  divisor  ;  but,  since  it  shows  the  number 
of  parts  into  which  the  whole  has  been  divided,  it  shows 
the  name  of  each  part,  and  is  therefore  called  the  denom- 
inator, 

The  upper  figure  shows  the  number  of  these  parts  taken, 
and  is  therefore  called  the  numerator, 

The  figure,  then,  above  the  line  denotes  number,  the  fig- 
ure below  the  line  name. 

106,  The  numerator  and   denominator  are   called  the 
terms  of  a  fraction. 


107,    A  proper  fraction  is  one  of  which  the  numerator  is 
less  than  the  denominator  ;  as  -. 


COMMON  FRACTIONS.  113 

108,  An  improper  fraction  is  one  of  which  the  numerator 
equals  or  exceeds  the  denominator  ;  as  -|,  *£-. 

When  the  numerator  is  greater  than  the  denominator,  more  than 
one  unit  must  be  regarded  as  divided  into  equal   parts  ;  thus,     f 

I      i      i      i      I      i      i      i      I      i      i      i      I 

means  that  three  units  have  been  divided  each  into  four  equal  parts, 
and  that  all  the  parts  of  two  units  and  one  part  of  the  third  unit  are 
taken. 

109,  A  mixed  number  is  an  expression  consisting-  of  a 
whole  number  and  a  fraction  ;  as  4-f-,  5.35.     These  expres- 
sions are  read  four  and  three-sevenths,  five  and  thirty-five 
hundredths. 

Every  mixed  number  means  that  some  entire  units  are  taken,  and 
the  fraction  of  another  unit. 

Select  the  proper  fractions,  the  improper  fractions,  and 
mixed  numbers  from  the  following  expressions  : 

f  ,  ¥.  H,  i.  9f  W.  i  if  ¥.  sf  , 

.  6f  ,  ^,  18f,  f. 


110,  An  improper  fraction  represents  a  quantity  which 
can  also  be  represented  by  a  whole  number  or  else  by  a 
mixed  number.     Thus,  -^  —  2f  . 

For,  if  we  suppose  several  units  to  he  divided  each  into  seven 
equal  parts,  and  we  take  19  of  these  parts,  14  (that  is,  2x7)  will 
make  2  units,  and  the  five  remaining  parts  will  be  five-sevenths  of 
another  unit. 

111,  To  reduce  an  improper  fraction  to  a  whole  or  mixed 
number, 

Divide  the  numerator  by  the  denominator. 

The  quotient  will  be  the  whole  number,  and  the  remainder,  if 
any,  will  be  the  numerator  of  the  fractional  part,  of  which  the  de- 
nominator is  the  same  as  the  denominator  of  the  improper  fraction. 


114  COMMON  FRACTIONS. 


Ex.  70.     (Oral.) 

Reduce 

to  whole 

or  mixed  numbers  : 

1.  J£- 

7. 

-V*.                  13.  f. 

19. 

¥• 

2.  ty. 

8. 

_2£.                  14.   2^ 

20. 

«- 

3-  ¥• 

9. 

¥•                  15-  if- 

21. 

¥• 

4-  «• 

10. 

J^.                  16.  ^l. 

22. 

tf 

5.  ^. 

11. 

14                          17  .    .3^0 

23. 

¥• 

6.  tt. 

12. 

V-                   18.  Y- 

24. 

Ex.  71. 
Reduce  to  whole  or  mixed  numbers : 

1.  fj.  9.  3^.  17.  ^.  25. 

2.  ^^..  10.  ^L.  18.  ^Z-.  26. 

3.  -^tV*  11.   AJM  19.  ^234  27. 

4.  ^3,  12.   ^£.  20.  -3452y.  28. 

5.  4^L.  13.  m.  21.  -HrfP*  29- 

6.  4.^..  14.   |!$.  22.  ^^2.  30. 

7.  ^..  15.    3^.  23.  ^-.  31. 

8.  300  16.   AJJ1.  24.  HP-.  32. 


112,  A  whole  number  or  a  mixed  number  represents  a 
quantity  which  can  also  be  represented  by  an  improper 
fraction.     Thus,  $ 3|  =  $J^. 

For  each  dollar  contains  4  fourths;  therefore  3  dollars  contain 
3  X  4  fourths  or  12  fourth*;  which,  together  with  the  3  fourths,  make 
15  fourths.  Hence, 

113,  To  reduce  a  mixed  number  to  an  improper  fraction, 
Multiply  the  whole  number  by  the  denominator  of  the 

fraction,  and  to  the  product  add  the  numerator ;  under  this 
sum  write  the  denominator. 


COMMON    FRACTIONS.  115 

114,     A  whole  number  may  be  expressed  as  a  fraction 
with  any  given  denominator.     Thus,  9  —  4^. 

For,  as  1  unit  contains  7  sevenths,  9  units  contain  9x7  sevenths, 
or  63  sevenths. 

A  whole  number  may  be  written  in  the  form  of  a  frac- 
tion with  1  for  a  denominator.     Thus,  9  =  -f-. 

Ex.   72.     (Oral.) 
Reduce  to  improper  fractions  :  * 


1.  4*. 

7.   12f 

13.   llf 

19.  25f 

2.   7f. 

8.   lOf 

14.   13TV 

20.  30|. 

3.  8f 

9.   7|. 

15.   7A- 

21.   17TV 

4.  Of 

10.  3^. 

16.  9&. 

22.   40£. 

5.  5f 

11.  8/T. 

17.  20-&. 

23.  50f 

6.  6f 

12-  3&. 

18.  15J. 

24.  80|. 

25.  Change  12  to  thirds;  8  to  fourths;  7  to  fifths;  9  to 
halves ;  12  to  ninths  ;  13  to  sixths ;  11  to  sevenths  ; 
14  to  eighths. 

Ex.  73. 
Change  to  improper  fractions : 


1.  15-^j. 

6.  45^-. 

1  1  .    5^2  7 

16.   155|^. 

2.  36H- 

7.  56&. 

12.  46^. 

17.  17||. 

3P\6  1 
.     U-Jr-A-. 

8.  77^. 

1  o      cy  7 

18.   167H. 

4<   318 

9.   183^-. 

14.   9f|. 

19.  29||. 

5.  12Jf  ,t 

10.   72H- 

15.   10^. 

20.  29ff. 

21.  Change  25  to  94ths;  218  to  23ds;  375  to  87ths. 


116  COMMON   FRACTIONS. 

REDUCTION  OF  FRACTIONS  TO  LOWER  TERMS. 

115,  If  the  numerator  and  denominator  of  a  fraction  be 
both  multiplied  or  both  divided  by  the  same  number,  the 
value  of  the  fraction  is  not  altered. 

J     i     i     I     i     i     I     i     i     I     i     i          t     i     If 

0      '  D  B  P 

Thus,  if  the  line  AB  be  divided  into  5  equal  parts  at 
the  points  (7,  D,  E}  and  F,  then  ^.Fis  £  of  AB. 

Now,  if  each  of  the  parts  be  sub-divided  into  3  equal 
parts,  AB  will  contain  15  of  these  sub-divisions,  and  AF 
12  of  these  sub-divisions.  Therefore  AF  is  -J-J  of  AB. 

Since  AFis  $  of  AB  and  also  -J-J  of  AB,  it  follows  that 
Y$  —  £.  But  •$•  is  obtained  from  -J-J  by  dividing  both 
numerator  and  denominator  by  3.  Therefore, 

To  reduce  a  fraction  to  lower  terms, 
Divide  the  numerator  and  denominator  by  any  common 
factor. 

A  fraction  is  expressed  in  its  lowest  terms  when  both  the  numer- 
ator and  denominator  are  divided  by  the  greatest  common  divisor. 

Eeduce  ff  £  to  its  lowest  terms. 

336  —    84    —  21  —  8 
T8T        TinT        flF        7' 

The  common  factors  cancelled  are  4,  4,  and  7. 

Reduce  |-f£  to  its  lowest  terms. 

Since  no  common  factor  can  readily  be  detected,  we  find  theG.C.M 

259)333(1 
259 
74)259(3 

222 
37)74(2 

74 

Divide  259  and  333  each  by  37,  their  G.C,M.     Then  f ff  =  f 


COMMON   FRACTIONS.  117 


Ex.  74.     (Oral.) 

Reduce  to  lowest  terms  by  inspection  : 

1. 

A- 

8. 

H- 

15.   f£. 

22. 

if 

2. 

& 

9. 

H 

16.   |f 

23. 

»• 

3. 

!i 

10. 

H- 

17.  if. 

24. 

If 

4. 

«• 

11. 

A- 

18.   if 

25. 

H. 

5. 

H 

12. 

A- 

19.  A. 

26. 

H« 

6. 

tt 

13. 

if- 

20.  |f 

27. 

H 

7. 

H. 

14. 

|f. 

21.  if. 

28. 

If. 

Ex.  75. 

Reduce  to  lowest  terms  by  the  method  of  inspection  or 

by  the  method  of  finding  the  G.C.M. 

!•  m-      i3-  AV      2s-  «*•  37. 

2.    ^  n  o  14     |  o  §  26     i  3  §  38 

3-  Vft-  lg-  Ht-  27.  T?^V  39. 

4.    1  ?  ^  16.    ^  1 7  28.    ^  f  %-  40 

5-  -^r-  17-  iff.  29.   fff.  41.   |ff. 

6.  iff  18.  |JJ.  30.  T^.  42.  ^/ft. 

7-  T^T-  19-  Iff  31.  Ml-  43.  ||f. 

8.   ||f.  20.   |f  32.  ||f.  44.  ^V. 

9-  -Ms-  21.  ^j..  33.  f||.  45.  iff. 

10-  t%-  22.  £f|.  34.   |^.  46.  |^|. 

11.  ^.  23.   ff|.  35.  iff  47. 

12.  i*4.  24.  m.  36.  444.  48. 


NOTE.    In  the  answers  to  all  examples,  fractions  should  be  left  in 
their  lowest  terms. 


118  COMMON   TRACTIONS. 

MULTIPLICATION  OF  FRACTIONS. 

116,  7x3  horses  =  21  horses. 
7  X  3  fifths   =  21  fifths. 

If  three  like  quantities  are  taken  7  times,  the  result  will 
be  7  times  3  of  the  same  quantities. 

f  X  15  means  £  of  15,  which  equals  9.     Hence, 

117,  To  find  the  product  of  a  whole  number  and  a 
fraction, 

Find  the  product  of  the  numerator  and  whole  number, 
and  divide  the  result  by  the  denominator. 

A  factor  common  to  the  whole  number  and  the  denominator  of 
the  fraction  may  be  cancelled.  For,  cancelling  a  factor  common  to 
the  whole  number  and  the  denominator  of  the  fraction  before  the 
multiplication,  is  evidently  equivalent  to  dividing  the  numerator 
and  denominator  of  the  resulting  fraction  by  that  factor  after  the 
multiplication.  Which  may  be  done  by  §  115. 

Ex.  76.  (Oral.) 
Find  the  products  of : 

1.  18  xf         10.  14  Xf  19.  Sxft.  28.  ISxfa. 

2.  25  X  f         11.  20  X  f  20.   16  X  fa.  29.  25  X  fa. 

3.  27  Xf         12.   16  Xf  21.  36X-&.  30.  fa  X  26. 

4.  10  Xf-       13.     7x£f  22-  24  Xf  31.  £x9. 

5.  24  X  f         14.  16  X  f .  23.  32  X  fa.  32.  T77  X  12. 
6-   12  X  f         15.  ft  X  10.  24.   12  X  fa.  33.  22  X  fa. 
7     21  X  f          16.  ft  X  12.  25.   27  X  f  34.  fa  X  28. 
•8.-30xf         17.  27  Xf  26.   18  Xf  35.  fa  X  21. 
9.    16  X  f          18.   ft  X  15.  27.    12  X  f  36.  fa  X  35. 


COMMON   FRACTIONS.  119 


118,    To  multiply  a  fraction  by  a  fraction. 
Multiply  -fby  f . 


I     .     ,     I     ,     .     IP 


|-  multiplied  by  -|  means  -|  of  f. 

If  the  line  AB  be   divided  into  5  equal  parts  at  the 
points  <7,  D,  E,  and  F,  AF  will  be  \  of  AB.     Now,  if 
each  part  be  sub-divided  into  three  equal  parts,  there  will 
evidently  be  15  such  parts  in  the  whole  line,  and  each  part  . 
will  be  ^5-  of  the  line. 

That  is,  ^  of  -J-  is  y1^  of  the  whole. 

J  of  f  will  be  ^  +  -^  +  ^4^,  or  ^  of  the  whole. 
And  |  of  f  will  be  twice  -fe  ;  that  is,  -^  of  the  whole. 
Therefore, 

To  multiply  a  fraction  by  a  fraction, 

Find  the  product  of  the  numerators  for  the  required 
numerator,  and  of  the  denominators  for  the  required  denom- 
inator. 

Mixed  numbers  must  first  be  reduced  to  improper  fractions. 
Any  factor  common  to  a  numerator  and  denominator  should  be 
cancelled  before  the  multiplication. 


(2)  tf 

Reducing  the  mixed  number  to  an  improper  fraction,  we 
have, 

H  x  f|  x  H. 

By  cancellation, 

,       2       1 


3 


120  COMMON   FRACTIONS. 


Ex.  77.     (Oral.) 
Find  the  products  of : 

I-    t  X  f .  4.    }  X  f .  7.   f  X  J.          10.   ^  X 

2.   1-  X  f .  5.   t  X  f  8.   ^  X  f       11.   i-  X  Y- 

3-    J  X  f  6.   TV  X  f          9.   ^  X  10.     12.   iJ  X  f 

Ex.  78. 
Find  the  products  of: 

1.  tX«.  18.  Si  of  A  of  A- 

2.  f  X4f  19.  -ft  of  2$  of  21. 

3.  2£  X  -ft  X  f  20.   J  of  l^j  of  3f 

4.  ^XS-ft-Xf  21.  8|Xfoff 

5.  |  of  H  of  If  22.   ICx^of^. 

6.  |  X  |  X  f  23.   3  X  7^  X  H  X  3^. 

7-  |fX|Xf  24.   A°ffX*  XHX77. 

8.   3§  X  6J  X  H.  25.  H  X  f  X  ft  X  J. 

9-  Hx9iX^.  26.  ^x^XifXii. 

10.  2£  X  f  X  &.  27.  3^  X  4J  X  15. 

11.  2f  of  f  of  -fa.  28.  T%  of  7  X  H  of  87^-. 

12.  2^of^of^f.  29.  llf  x  16^r  of  ^  of 

13.  f  X  7*  X  A-  30.   ^  of  |  of  2|  X  15f 

14.  |x7|xA-  81.  8^  of  AX  2}  off 

15.  2fc  of  3$  of  ^.  32.  3|  of  f^  of  6|  of  f£. 

16.  ^  of  T«T  of  7f  33.   I  X  2J  X  4^  X  A- 

17.  3|  of  |  of  H.  34.   f  X  2f  X  7|  X  4t  X  f 


COMMON   FRACTIONS.  121 


35.  2ix£xfx3fx5i.       38.    ^ 

36.  |  X  |  X  7£  X  5J  X  6f       39.   3|  of  -ft  of  £  of  10. 

37.  51  X  ff  X  7£  X  M  X  7J.  40.   ft  X  21  X  3f  X  2J  X  ft. 

119,  When  the  product  of  a  mixed  number  and  a  whole 
number  is  required,  it  is  generally  best  to  find  the  product 
of  the  whole  number  and  the  fractional  part  of  the  mixed 
number,  then  the  product  of  the  whole  number  and  the 
integral  part  of  the  mixed  number,  and  combine  the  results. 
Thus, 

The  product  of  9  times  7£  is  found  as  follows  : 

OPERATION. 

1 

64| 

Here  9  times  J  equals  1  J,  the  J  is  written,  and  the  1  is  carried  to 
the  product  of  9  x  7,  making  64. 

Ex.  79. 
Find  the  products  of  : 


1. 

3x4f 

11. 

5f 

X9. 

21. 

20  x  5£. 

2. 

5x7|. 

12. 

2^ 

frXl5. 

22. 

4f  x  17. 

3. 

21  x  18f 

13. 

2f 

^X20. 

23. 

5|  x  18. 

4. 

22  X  29ft. 

14. 

H 

Xl2. 

24. 

6|  x  15. 

5. 

25  x  12f  . 

15. 

n 

X8. 

25. 

9|  X  21. 

6. 

6x2f 

16. 

6f 

X9. 

26. 

10J-  X  41. 

7. 

7x2f. 

17. 

9x3f. 

27. 

11|  x  32. 

8. 

8x2f 

18. 

12 

X2J. 

28. 

15f  X  36. 

9. 

6|x9. 

19. 

13 

x3f 

29. 

16£x40. 

10. 

3ft  x  10. 

20. 

16 

x  9f 

30. 

13|  x  27. 

122 


COMMON   FRACTIONS. 


DIVISION  OF  FRACTIONS. 

120,  When  a  product  of  two  numbers  is  equal  to  1,  each 
of  these  two  numbers  is  called  the  reciprocal  of  the  other. 

Thus,  5  x  J  =  1.  Hence,  the  reciprocal  of  J  is  5,  and  the  recipro- 
cal of  5  is  £.  Again,  j  x  f  =  1.  Therefore,  the  reciprocal  of  £  is  $, 
and  the  reciprocal  of  ^  is  J. 

121,  To   multiply   by   the   reciprocal  of  a   number   is 
the  same  as  to  divide  by  the  number. 

Thus,  to  multiply  by  J,  means  to  separate  the  multiplicand  into 
three  equal  parts,  and  to  take  one  of  the  parts  for  the  required  pro- 
duct ;  and,  to  divide  by  3  means  to  separate  the  dividend  into  three 
equal  parts,  and  to  take  one  of  the  parts  for  the  required  quotient. 

To  multiply  by  §  means  to  separate  the  multiplicand  into  three 
equal  parts,  and  to  take  two  of  these  parts  ;  and  to  divide  by  J,  the 
reciprocal  of  §,  means  to  divide  the  dividend  into  three  equal  parts. 
and  to  take  two  of  these  parts. 

Hence,  to  divide  by  a  whole  number  or  a  fraction, 

Multiply  by  its  reciprocal. 

Thus,  *-*-2  =  ixf  =  A- 


Mixed  numbers  must  first  be  reduced  to  improper  fractions. 

Ex.  8O.     (Oral.) 
Find  the  quotients  of : 


2.  ftn-4. 

3.  f  -r-2. 

4.  ft  -3. 

5.  i^-f-2. 

6.  f-5. 


8.  H- 

9.  ff--f- 

10.  A+ 

11.  I 
12. 


ia.  -^--6. 

14.  -^--3. 

15.  ff--6. 

16.  ^ -3. 

17.  ft -4. 


19. 
20. 
21. 
22. 
23. 


18.    ft -12.       24.   -8--«- 


COMMON   FRACTIONS.  123 

Ex.  81. 
Find  the  quotients  of : 

1.  J|-12.  12.  f^7TV  23.  f  ^lf- 

2.  ff^25.  13.3^21.  24.^  +  ^. 

3.  M-12-  14-   l  +  f  25.  3ii^fr 

4.  ft  H- 13.  15.  f  *f  26.  3|^9f 

5.  tf$-f.  19.  16.  ioffn-f  27.   9£  -+-  3f 

6.  AH- TV  17-   ioff-A-  28.   8^^. 

7.  I-*-!.  18.  iofSjH-H.  29.  19+1J. 

8.  2J  +  J.  19.  if-f-jf  30.  3f^3f 

9-   i+f  20.   Jl-^f  31.  ioff-frA- 

10.  |-n2f  21.  8|H-6|.  32.   Ijof7i^-2f. 

11.  Gl-4f  22.   7|^8|.  33.  2Joflf-H2J. 

34.  1£  -H  ^.  of  |.  40.  3|  of  5J  of  7|  ^-  63. 

35.  f  of  2J  -:-  H  of  6f  41.  3}  of  7|  of  If  -  Si. 

36.  |- of  4£  H- -J  of  3f.  42.   ?iof  S^H-l^of  If 

37.  2|  of  1^  -*•  5i  of  3|.  43.  9  H-  ^  of  1^.  of  4|. 

38.  2iof2i^f¥of3|.  44.  lC 

39.  1'i-s-liof  A-of  A-  45.  3f  of 

122,  When  a  mixed  number  is  to  be  divided  by  a  whole 
number,  it  is  best  to  divide  the  integral  part  of  the  dividend 
first,  and  then  the  fractional  part.  If  there  is  a  remainder 
from  dividing  the  integral  part,  this  remainder  may  be 
put  with  the  fraction,  and  the  result  reduced  to  an  improper 
fraction,  and  then  divided  by  the  divisor. 


124  COMMON   FRACTIONS. 

Divide  16|-  by  4  ;  16$  by  7. 

OPERATION.  OPERATION. 


In  the  first  problem  we  simply  divide  the  whole  number  16  by  4, 
and  then  the  fraction  |  by  4,  and  obtain  the  result  at  once,  4J. 

In  the  second  problem  we  divide  the  16  by  7,  and  obtain  the 
quotient  2  and  a  remainder  2.  The  remainder  is  joined  with  the  f  , 
making  2J  =  f  ,  and  J  -j-  7  -  /p 

Ex.  82. 
Find  tbe  quotients  of  : 


1.    19$-*-  3. 

6.   34$-*-  17. 

11.   65^-^-9. 

2.    12$-*-  5. 

7.    31$  +-11. 

12.    147$  -f-  13. 

3.    24§-f-8. 

8.    371-7-18. 

13.    76$-*-  19. 

4.    19f-6. 

9.   45^+7. 

14.    124^-^6. 

5.    17$-*-  9. 

10.   57|-*-  16. 

15.   326^15. 

Ex.  83. 

1.  What  must  be  paid  for  24  yards  of  cloth,  at  $f  per 

yard? 

2.  A  farmer  bought  327  sheep,  at  $4§  a  head  ;  required 

the  cost  of  the  flock. 

3.  At  25  cents  a  pound,  what  must  be  paid  for  82J  pounds 

of  butter? 

4.  A  merchant  sold  15£  yards  of  silk,  at  $4S  per  yard  ; 

what  change  should  he  give  back  from  8  ten-dollar 
bills? 

5.  If  beefsteak  cost  22  cents  per  pound,  and  mutton  chops 

21  cents,  how  much  will  a  man  pay  for  meat,  who 
eats  J  pound  of  beefsteak  for  breakfast,  and  li 
pounds  of  mutton  chops  for  dinnpr  ? 


COMMON   FRACTIONS.  125 

6.  At  $|-  per  yard,  how  much  cloth  can  be  bought  for 

$25? 

7.  If  $19}  be  paid  for  9  yards  of  silk,  what  is  the  cost 

per  yard  ? 

8.  A  man  walks  3 7^  miles  in  6  hours ;  how  many  miles 

does  he  walk  an  hour  ? 

9.  A  farmer  sells  19}  acres  of  land  for  $375  ;  what  is  the 

price  per  acre  ? 

10     A  lady  pays  $3  for  -|  of  a  yard  of  silk ;  what  is  the 
price  per  yard  ? 

11.  A  man  in  one  year  pays  $45.26  for  cigars,  the  average 

price  of  which  is  6-J-  cents  apiece.  How  many  does 
he  smoke  in  a  year  ? 

12.  If  -£  of  an  acre  of  tillage,  land  cost  $125,  what  is  the 

price  per  acre?  How  many  acres  can  be  bought 
for  $1297? 

13.  Gideon  Lyford  earns  $30  per  week  ;  what  will  remain 

at  the  close  of  the  week  when  he  has  paid  for  6§ 
pounds  of  butter,  at  33  cents  a  pound,  10}  pounds 
of  mutton,  at  20  cents  a  pound,  8$  pounds  of  beef, 
at  25  cents,  3  boxes  of  strawberries,  at  16  cents,  150 
pounds  of  ice,  at  }  cent,  20  loaves  of  bread,  at  10 
cents,  fuel  $2,  vegetables  $3? 

14.  Find  the  product  of  17f  X  8£  of  6^. 

15.  If  a  man  build  ^  of  a   rod  of  wall  in  one  hour,  how 

much  will  he  build  in  •£•  of  an  hour? 

16.  If  a  ship  costs  $16,785,  what  will  f  of  it  be  worth  ? 

17.  If  a  water-pipe  discharges  16  J  barrels  of  water  in  an 

hour,  how  many  barrels  will  it  discharge  in  9^-  hours? 

18.  For  4  sheep  $25f  are  paid;    what  is  the   price  per 

head? 


126  COMMON   FRACTIONS. 

19.  A  coal  dealer  paid  $375  freight  for  transporting  coal 

from  Scranton  to  Hudson.     If  the  price  was  $£  per 
ton,  how  many  tons  were  transported  ? 

20.  How  many  pounds  of  beef,  at  18}  cents  per  pound, 

can  be  bought  for  $17.48? 

21.  A  farmer  hires  an  equal  number  of  men  and  boys,  and 

pays  for  a  man  and  boy  $2f  a  day.     If  the  pay  roll 
is  $84  a  day,  how  many  men  and  boys  does  he  hire  ? 

22.  When  Spff^  acres  of  land  cost  $1297,  what  will  £  of 

an  acre  cost? 

23.  A  city  speculator  in  land  divided  }  of  an  acre  into  lots 

of  -jJg-  of  an  acre  each,  and  sold  them  all  for  $  13,426f . 
What  was  the  average  price  per  lot  ? 

24.  For  §  of  J  of  a  ship  the  sum  of  $6394  was  received ; 

what  is  the  value  of  the  ship  ? 

25.  A  vessel  sails  17$  miles  per  hour;  how  many  miles 

will  she  sail  in  26$  hours? 

26.  George  is  13 1  years  old,  Henry  is  }  as  old  as  George, 

and  John's  age  is  1-J  that  of  Henry  ;  what  is  the 
age  of  John  ? 

27.  There  are  16J  feet  in  one  rod;    how  many  feet  are 

there  in  84|-  rods? 

28.  How  many  feet  around  a  field,  each  one  of  whose  four 

sides  measures  7-|  rods  ? 

29.  A  schooner  sails  on  the  average   175-|  miles  a  day; 

how  far  will  she  sail  in  a  week  ? 

30.  At  the  rate  of  8f  miles  per  hour,  how  many  miles  will 

a  ship  sail  from  a  quarter  past  three    A.M.    to   a 
quarter  before  six  P.M.  ? 

31.  Reduce  f  of  ^  of  -^  of  |-|  to  a  simple  fraction  in  its 

lowest  terms. 


COMMON   FRACTIONS.  127 

32.  George  Ward  inherited  from  his  father  -J-J-  of  a  farm 

containing  377  acres.  He  divided  his  share  equally 
among  his  four  sons ;  how  many  acres  would  each 
one  of  the  sons  receive  ? 

33.  How  many  pounds  of  sugar,  at  9}  cents  a  pound,  can 

be  bought  for  $1.52? 

34.  For  231  baskets  of  peaches,  a  grocer  gave  $20.59; 

what  was  the  price  per  basket  ? 

35.  A  farmer  sold  42  bushels  of  potatoes  for  $26.58 ;  what 

was  the  average  price  per  bushel  ? 

36.  At  37}  cents  per  yard,  how  many  yards  of  lace  can  be 

bought  for  $5i? 

37.  A  farmer  sold  6}  bushels  of  apples  for  $4.87}  ;  what 

was  the  price  per  bushel  ? 

38.  When  6}  bushels  of  apples  bring  $3.90,  what  are  they 

worth  a  peck  ?     (Four  pecks  make  a  bushel.) 

39.  At  60  cents  a  pound,  how  many  pounds  of  tea  can  be 

bought  for  $4.65? 

40.  If  i^j-  of  a  yard  of  cloth  cost  80  cents,  what  should  be 

paid  for  -j^-  of  a  yard  ? 

41.  The  cost  of  fencing  a  lot  8}  rods  in  circuit  is  $6£, 

what  is  the  rate  per  rod  ? 

42.  A  roll  of  carpeting  containing  202  yards  is  cut  into 

pieces  of  25 1  yards  each,  and  each  piece  is  sold  for 
$32}.  Required  the  number  of  pieces,  and  the 
price  per  yard. 

43.  When  35}  bushels  of  turnips  cost  $28.60,  what  should 

be  paid  for  }  of  a  bushel  ? 

44.  How  many  yards  of  cloth  can  be  bought  for  $10.80, 

if  •£$  of  a  yard  cost  63  cents  ? 


128 


COMMON   FRACTIONS. 


LEAST  COMMON  DENOMINATOR. 

123.  A  fraction  is  changed  to  higher  terms  by  multipli- 
cation. 

Reduce  £  to  twelfths. 
Multiply  both  terms  by  3  ;  thus, 

3x8      24 
3x4=12 

In  either  of  the  two  forms  f  or  f  J  the  value  of  the  fraction  is  2. 

124.  Hence,  to  reduce  a  fraction  to  higher  terms, 

Multiply  both  terms  of  the  fraction  by  that  number  which 
will  change  the  given  denominator  to  the  required  denomi- 
nator. 

The  required  multiplier  is  found  by  dividing  the  required  denom- 
inator by  the  denominator  of  the  given  fraction. 


Reduce  : 


Ex.  84.    (Oral.) 


1. 

|  to 

20ths. 

8. 

f  to 

27ths. 

15. 

fV 

to 

26ths. 

2. 

H° 

lOths. 

9. 

rr  to 

33ds. 

16. 

TV 

to 

36ths. 

3. 

I  to 

9ths. 

10. 

^  to 

28ths. 

17. 

f 

to 

Slsts. 

4. 

^to 

14ths. 

11. 

A  to 

36ths. 

18. 

A 

to 

96ths. 

5. 

|  to 

18ths. 

12. 

•ft  to 

20ths. 

19. 

•& 

to 

44ths. 

6. 

f  to 

12ths. 

13. 

•fa  to 

45ths. 

20. 

1 

to 

16ths. 

7.  f  to  24ths.         14.  -2^  to  lOOths.     21.  ^  to  72ds. 

125,  Similar  fractions  are  fractions  that  have  a  common 
denominator. 

All  fractions  must  be  expressed  as  similar  fractions  before  they 
can  be  added  or  subtracted,  and  in  all  cases  it  is  best  to  express  them 
with  the  least  common  denominator.  (L.C.D.) 


COMMON   FRACTIONS.  129 

126.  The  least  common  denominator  of  two  or  more 
fractions  is  the  least  common  multiple  of  their  denominators. 

Reduce  -J-,  J,  -J-  to  similar  fractions. 

The  least  common  multiple  of  the  denominators  2,  3,  4  is  12.  It 
is  therefore  necessary  to  reduce  J,  J,  J  to  12ths,  by  the  method  ex- 
plained in  \  124,  and  we  have  ^,  T\,  -fa. 

127,  Hence,  to  reduce  fractions  to  similar  fractions, 
Find  the  least  common  multiple  of  the  denominators ; 

this  will  be  the  required  denominator.  Divide  this  denomi- 
nator by  the  denominator  of  each  fraction. 

Multiply  the  first  numerator  by  the  first  quotient,  the  sec- 
ond numerator  by  the  second  quotient,  and  so  on. 

The  products  will  be  the  numerators  of  the  equivalent 
fractions. 

Ex.   85.     (Oral.) 
Reduce  to  similar  fractions  : 

1-  i,  i            5.  i,  f               9.  i,  *,  A-  13.  |,  f,  J. 

2-  *.  i             6-   i  f             10.  |,  |,  TV  14.  f,  |,  ft. 

3-  i,  i             7.   f  f.             11.   f  £,  f .  15.  |,  f  A. 
4.   i  f             8.  |,  f.             12.   f  f  f  16.  f  f  f 

Ex.  86. 
Keduce  to  similar  fractions  : 

1-  if,  If,  T^r-  6.  |,  ^  if,  ff. 

2-  A,  A-  H-  7.  f  f ,-A,  A- 

3-  A.  A-  f •  8-  i,  f  if-  «,  AT- 

4-  A,  A-  f  9-  t,  A,  «,  If  -  A- 

5-  f  A.  A-  10.  |,  f,  f  A,  if 


130  COMMON  FRACTIONS. 

ADDITION  OF  FRACTIONS. 

128.  Add  |,  f,f 

These  fractions  reduced  to  similar  fractions  become  ^y,  y9^,  T£,  and 
A  +  ft  +  ii-H-2A-2J-  2J.  An*. 

129.  Hence,  to  add  fractions, 

Reduce  the  fractions  to  similar  fractions,  and  write  the 
sum  of  the  numerators  over  the  common  denominator. 

Add  |,  A,  A- 

~    8     12     15 


6     15 


2      3     15 


2      1      5 
Hence,  L.  C.  D.  -  2s  x  3  x  5  =»  120. 

Numerators  .  .  . 

Sum  of  numerators  = 
Therefore,  sum  of  fractions  =  f  $  =  J$  =  IfJ.  Iff  Ans. 

130,  If  any  of  the  expressions  are  integers  or  mixed 
numbers,  add  together  separately  the  integers  and  the  frac- 
tions, and  find  the  sum  of  the  results. 

Find  the  sum  of  2&,  !•&,  5|£. 

L.  C.  D.  of  the  fractions  =  22  X  3  x  5  =  60. 

f    9 

Numerators  .  .  .  .  <  28 

I « 

Sum  of  numerators  =  92 

Sum  of  fractions     =  f  f  =  f  |    -  1^ 

Sum  of  integers      =  2  +  1  +  5  =  8 

9A.  Ans. 
Ex.  87.     (Oral.) 
Find  the  sum  of : 

i-  A>  A-       3-  !>  f         5-  tt.  A-      7-  7i  3i 

2-  A,  A-         4.  A,  A-          6.  ft,  «•          8.  8|,  4f 


COMMON  FRACTIONS.  131 

9.   5f ,  4f  13.   f ,  |.  IT.   i,  A.  21.  8ft,  7f 

10.  9fV,  5TV  14.  |,  |.  18.  |,  ||.  22.  6f ,  5J. 

11.  8f,  5f.  15.  -1%,  f.  19.  SjV,  4A-  23.  7f,  4f 

12.  TTV,  3iV  16.  |,  ^.  20.  2-f,  8f.  24.  Oft,  8f 

Ex.  88. 
Find  the  sum  of: 

1-  i,  i  f  13-  A,  f,  f  ft- 

2-  i  A-  f  14.  f ,  |,  H,  'TV- 
S' I.  f.  i  15.  -fr,  A,  H.  f 

4-  I  f,  f  16-  H.  M-  i  T3?- 

5-  f  f  f  17.  i  A-  A.  A- 

6-  I  i  «•  18-  A.  A-  «•  H- 

7.  4f  3|,  6A-  19-  I  i,  f,  I,  I  H- 

8.  7|,  8H,  9|.  20.  |,  A,  f,  ||,  A,  «• 
»•  8A.  4|.  S|.                       21.  |,  A,  H,  If,  «,  «• 

10.  4|,  5|,  6|.  22.  H,  «,  A.  if,  A-  fi- 

ll. 9f,  4f,  8f.  23.  |f,  tf.  If,  «,  i|,  «. 

12.  4|f,  5|f,  6^.  24.  |,  |,  ^,  ff,  «,  |i. 

25.  24|,  13f  36|,  60|.  47f. 

26.  35|,  17f  25f,  48|,  18J. 

27.  54|,  28|,  16|,  36f,  64|. 

28.  36f  87f  59f,  54f  16|. 

29.  23f,  32f,  18f,  27|,  28|. 

30.  74^,  64H,  4S&,  2SH, 


132  COMMON   FRACTIONS. 


SUBTRACTION  OF  FRACTIONS. 

131.    From  |£  take  -ft. 

24  =  23  x  3, 
18  =  2  X  32. 

Hence,  die  L.C.D.  =  23  X  32  =  72. 


132,  Hence,  to  subtract  one  fraction  from  another, 

Reduce  the  fractions  to  similar  fractions. 
Subtract  the  numerator  of  the  subtrahend  from  the  numer- 
ator of  the  minuend. 

Write  the  result  over  the  common  denominator. 

133,  If  the  terms  are  mixed  numbers,  subtract  separately 
the  integers  and  fractions,  and  unite  the  results. 

Subtract  5f  from  15f  . 
Here  theL.C.D.  =  8. 

15f  -  5f  =  10*^  =  10f  10f  Ans. 


Subtract  3f  from 

=  Hi 


Ans. 

The  difference  between  5T\  and  3|  is  2^-°^--     Since  if  cannot  be 
subtracted  from  JJ,  1  is  taken  from  2,  and  added  to  JJ,  making  f  J. 

From  9  take  ||. 


COMMON   FRACTIONS.  133 


Ex 

89. 

Find  the  value  of  : 

i-  H-A. 

25. 

14- 

iV- 

49. 

24^-16|i. 

3.  n-n- 

26. 

21- 

if- 

50. 

92|  -  73f  . 

3.  ii  -M- 

27. 

20- 

H 

51. 

19T\-14|f. 

4-   fi-4i 

28. 

42  

tt 

52. 

23|  -  16f. 

5-  tt-W- 

29. 

25  -|f. 

53. 

15|-12|. 

6.   $-f. 

30. 

21- 

tt- 

54. 

42|-14|. 

7.  f-f. 

31. 

14- 

If. 

55. 

24-^-  -  15f  . 

84         1 
*    "5"       IT* 

32. 

13- 

A 

56. 

72f-28f 

9-  A  -A- 

33. 

24- 

13f. 

57. 

19|-13|. 

10.  |-f. 

34. 

42- 

15ft. 

58. 

26f-19|. 

11.  |i  —  f. 

35. 

20- 

1%. 

59. 

45^-26ft. 

12.   f-f. 

36. 

84- 

37|f 

60. 

34f-16f. 

1  Q        8             5 

1O.       ^-   T^J-. 

37. 

21A 

-11- 

61. 

34|-18f. 

14-  A  -A- 

38. 

27f-l. 

62. 

64f-28|f. 

15.  H-A. 

39. 

42T\ 

-A- 

63. 

48f-19f 

16-  M-li- 

40. 

26^ 

-H 

64. 

76|-72ft. 

i  n     2       i 
"5"      "S"* 

41. 

43li 

-H- 

65. 

97|-32||. 

is.  tf-f 

42. 

27A 

-if- 

66. 

90^  -9f. 

19.   If  ~|f 

43. 

»1A 

-f 

67. 

78ft  -56f. 

20.   H-^. 

44. 

3-H 

-'Jl- 

68. 

96|  -  49|. 

QI       l  9  2  3 

45. 

83ft 

1  3 
T6"' 

69. 

47|  _  43||. 

oo        l  1    1  3 

46. 

26ft 

-A- 

70. 

55f-54f 

23.   |f  —  T\. 

47. 

74A 

-ri 

71. 

69ft  -67f| 

24-  U-H- 

48. 

68|- 

-f 

72. 

69^-23|J. 

134  COMMON   FRACTIONS. 

Ex.    90. 

1.  A  country  merchant  received  on  Monday  $25f,  on 

Tuesday  $19J,  on  Wednesday  $231,  on  Thursday 
$32f,  on  Friday  $29J,  on  Saturday  $37£.  What 
had  he  left  after  paying  a  freight  bill  of  $19f,  and 
to  his  clerk  $  12 J? 

2.  A  farmer  sold  two  loads  of  hay,  one  for  $13i  and  the 

other  for  $16f ,  and  received  $25  down.  How  much 
is  still  due  ? 

3.  A  miner  digs  17$,  19^,  18f  ounces  of  gold.     In  wash- 

ing there  is  a  loss  of  3$  ounces.  How  much  gold 
has  he  left? 

4.  Henry  Cameron  had  three  wheat-fields;  the  first  pro- 

duced 217f  bushels,  the  second  309f ,  the  third  419$. 
He  sent  516$  bushels  to  a  flour  mill,  and  sold  193 
bushels.  How  many  bushels  had  he  left  ? 

5.  From  a  piece  of  cloth  containing  47-J-  yards,  22$  yards 

were  sold,  and  then  5|  yards  were  sold.  How  many 
yards  remained  ? 

6.  A  grocer  sold  2|  pounds  of  tea  to  one  man,  li  pounds 

more  to  a  second  man  than  to  the  first,  and  to  a 
third  man  1}  pounds  less  than  the  amount  he  sold 
the  first  and  second  together.  How  many  pounds 
did  he  sell  to  the  second  man,  and  how  many  to 
the  third  man  ? 

7.  Of  the  prismatic  spectrum  red  occupies  $,  orange  1%, 

and  yellow  -^.  What  part  of  the  whole  do  these 
three  colors  together  occupy  ? 

8.  What  part  of  a  piece  of  cloth  has  a  merchant  sold,  who 

has  cut  off  and  sold  -f$,  ^,  -fo,  and  ^  of  it  ? 


COMMON    FRACTIONS.  135 

9.    A  treasurer  has  expended  £-£-,  ^-,  ^f,  -£-$,  and  -£$  of  a 

given  sum.     What  part  of  the  whole  has  he  left  ? 

• 

10.  Of  a  pole  %  is  blue,  -f-  red,  and  the  rest  white.     What 

part  of  it  is  white  ? 

11.  A  jeweller  has  used  -j^-,  -^,  and  -^  of  an  ingot  of  gold. 

What  part  of  it  still  remains  ? 

12.  A  student  has  read  -j5T,  ^,  and  £  of  a  certain  book. 

What  part  of  it  has  he  yet  to  read  ? 

13.  A  traveller  has  gone  -J-  of  a  journey  on  foot,  ^  on 

horseback,  ^  by  rail,  and  the  rest  by  coach.  What 
part  of  the  journey  has  he  gone  by  coach? 

14.  Of  the  component  elements  of  albumen  ^  is  carbon, 

y^  hydrogen,  and  •£%  nitrogen.  What  part  of  the 
whole  do  these  elements  constitute? 

15.  Add  together  the  greatest  and  least  of  the  fractions, 

f ,  £,  -J-J-,  ££,  and  subtract  this  sum  from  the  sum  of 
the  other  two  fractions. 

16.  How  many  tons  of  ore  must  be  raised  from  a  mine  so 

that,  on  losing  ^J  in  roasting,  and  -fe  of  the  remain- 
der in  smelting,  there  may  be  obtained  506  tons  of 
pure  metal? 

17.  A  man  invested  $  of  his  capital  in  bank  stock,  f  of  the 

remainder  in  real  estate,  and  had  left  $6000.  Find 
his  capital. 

18.  A  man  invests  $  of  his  money  in  land,  £  in  bank  stock, 

-J-  in  railroad  stock,  and  has  $8000  left.  What  is 
his  fortune  ? 

19.  A  owns  •§•  of  a  ship,  and  B  the  remainder;  and  f  of 

the  difference  between  their  shares  is  $1500,  What 
is  the  value  of  the  ship  ? 


136  COMMON   FRACTIONS. 

COMPLEX  FRACTIONS. 

134.  The  quotient  f-^-f  may  be  written   in  the  form 

$,  in  which  the  dividend  is  the  numerator,  and  the  divisor 

I 

the  denominator  of  a  complex  fraction. 

135.  A  complex  fraction   has  a  fraction   in   either  its 
numerator  or  denominator,  or  in  both  of  them. 

The  reduction  of  complex  to  simple  fractions  is  similar  to  division 
of  fractions. 

Reduce  t  to  a  simple  fraction. 

Multiply  both  terms  by  4  and  we  have  -fc. 

Reduce  |  to  a  simple  fraction. 

D 

Multiply  both  terms  by  12,  which  is  the  L.C.  M.  of  4  and 
6,  and  we  have  at  once  •£$. 

Reduce  — *  to  a  simple  fraction. 
^v 

Multiply  both  terms  by  12  and  we  have  |ff  =  ^f . 

Ex.  91. 
Reduce  to  simple  fractions  : 

••4     «-5j-    5-f     '•£• 

7  11  1  10 

2.    — .  4.    —  6.    I.  8.    It 

llf  13*  *  H 


COMMON   FRACTIONS.  137 


9. 

I 

14.    ?i                      19.      ^°f  $  . 

it 

A 

10. 

tt 

QQ  7                                                  2 

If 

f 

11. 

H. 

lfi     f  of  3^-              20     | 

12. 

5f 

17    t  of  18*, 

4  of  7-J              21       . 

13. 

H 

19f 

1ft      ^y  Of  12f 

.                 ^r 

3  of    ^ 

136,    To  express  one  number  as  a  fraction  of  another. 

What  fraction  of  8  is  5? 

Since  1  =  J  of  8, 

5  =  5  x  J  of  8. 
That  is,  5-|  of  8. 

The  number  which  follows  "  of"  is  the  denominator,  and  the  other 
number  the  numerator  of  the  required  fraction. 

Ex.  92. 
What  fraction  of : 

1.  8  is  7?  7.  2|is|?  13.  3f  isf? 

2.  7  is  8?  8.  |  is  4£?  14.  5Jis4f? 

3.  6  is  2?  9.  2fisli?  15.  llf  is5|? 

4.  5  is  3?  10.  2£is5f?  16. 

5.  7  is  15?  11.  2|is5±?  17. 

6.  15  is  7?  12.  5Jis2t?  18    14fis4£? 


138 


COMMON   FRACTIONS. 


19.  7f  is  2|? 

20.  7|is|f  ? 

21.  |  of  10£isf|? 

23.  f  ofl2fisfi? 

24.  f  of  3|is  Jf? 

137.    To  reduce  a  decimal  to  a  common  fraction. 
Reduce  0.25  to  a  common  fraction. 
0.25  =  ,&&.  =  i 


25.  H°f 

26.  ^  is  i 

27.  £  is£of  2£? 

28.  33  is  2 J  of 

29.  27|f  is  2££  of 

30.  36  is  3f  of 


138,    Hence,  to  reduce  a  decimal  to  a  common  fraction, 

TFh'fe  the  figures  of  the  decimal  for  the  numerator ;  and 
1 ,  with  as  many  zeros  as  there  are  figures  in  the  decimal, 
for  the  denominator. 

Ex.  93. 

Reduce  to  common  fractions  : 


1. 

0.5. 

9. 

0.015. 

17. 

0.7168. 

25. 

1.6125. 

2. 

0.06. 

10. 

0.18. 

18. 

3.02. 

26. 

8.0396. 

3. 

0.15. 

11. 

0.125. 

19. 

5.85. 

27. 

2.18375. 

4. 

0.025. 

12. 

0.004. 

20. 

7.075. 

28. 

1.0725. 

5. 

0.7. 

13. 

0.032. 

21. 

0.15625. 

29. 

22.848. 

6. 

0.19. 

14. 

0.3125. 

22. 

0.46875. 

30. 

1.30125, 

7. 

0.135. 

15. 

0.0625. 

23. 

0.00256. 

31. 

17.875. 

8. 

0.005. 

16. 

0.0425. 

24. 

0.00375. 

32. 

2.9375, 

COMMON   FRACTIONS.  139 

139,  To  reduce  a  common  fraction  to  a  decimal. 
Change  1  to  a  decimal. 

8)3.000 
0.375 

140,  Hence,  to  reduce  a  common  fraction  to  a  decimal, 
Divide  the  numerator  by  the  denominator. 

141,  If  a  fraction,  when  reduced  to  its  lowest  terms,  con- 
tains in  the  denominator  any  other  factor  than  2  or  5  (the 
prime  factors  of  10),  the  division  of  the  numerator  by  the 
denominator  will  not  terminate.      In  general,  it  will  be 
sufficient  to  obtain  five  decimal    places  in  the  quotient. 
But  the  number  in  the  fifth  place  of  the  quotient  must 
be  increased  by  1  if  the  number  in  the  next  place  of  the 
quotient  is  five,  or  greater  than  five. 


Ex.  04. 
Change  the  following  fractions  to  decimals  : 

1.  £f.  5.    llff.  9.   If  13. 

2.  dft.  6.   y.  10.   £  14. 

3-  Tttr-  7-  Tffcr-  n-  H-  15- 

4.  1G.  8.  5.  12.  ii. 


Express  the  following  as  decimals  to  five  places  : 

16.  f.  19.  TV  22.   if-  25. 

17.  f  20.   if.  23.   iff.  26. 

18.  f.  21.  f  24.  jff.  27. 


140  COMMON   FRACTIONS. 


Ex.  95. 

Solve  the  following  problems,  first  changing  the  common 
fractions  to  decimals : 

1.  A  person  owed  $24,560.     When  he  has  paid  $8345^, 

$7234^,  $6472^,  how  much  does  he  still  owe? 

2.  A  man  sold  46|£  acres  of  land,  at  the  rate  of  $9i  an 

acre,  and  54 J  acres  at  the  rate  of  $2^-J.  How  much 
did  he  receive  for  the  whole  ? 

3.  A  merchant  purchases  346  pieces  of  cloth,  each  con- 

taining 32f  yards,  at  $1J  a  yard,  and  sells  the 
whole  for  $2^  a  yard.  What  does  he  gain  ? 

4.  A  merchant  purchased  8  yards  of  cloth  at  $6i  a  yard. 

What  sum  will  he  gain  per  yard  if  he  sells  the  whole 
piece  for$56f$? 

5.  A  man  bought  a  piece  of  land  for  $1046}  at  the  rate 

of  $  15  J  an  acre.  He  sells  it  for  $  17  £  an  acre.  How 
much  does  he  gain  on  the  whole? 

6.  A  merchant  purchased  15  casks  of  wine  of  25  gallons 

each.  He  paid  $980  for  the  wine,  $78J  tax,  $33f 
for  transportation.  He  sold  it  for  $3£  a  gallon. 
How  much  did  he  gain  ? 

7.  A  speculator  purchased  738  acres  of  land  for  $21,294. 

He  sells  -|  of  his  land  at  the  rate  of  $34f  an  acre, 
and  the  rest  at  the  rate  of  $35  per  acre.  What  does 
he  gain? 

8.  A  piece  of  cloth  is  29f  yards  in  length.     How  many 

pieces,  each  containing  Iff  yards,  can  be  cut  from 
it? 


COMMON   FRACTIONS.  141 

9.    How  many  postage-stamps,  each  containing  £f   of  a 
square  inch,  are  in  a  sheet  of  172£  square  inches? 

10.  Of  a  boat  worth  $5600,  A,  who  has  |£,  sells  f  of  nis 

share  to  B,  and  B  sells  £  of  his  share  to  C.  Find 
the  value  of  C's  share. 

11.  From  Montreal  to  Toronto,  by  the  Grand  Trunk  Rail- 

way, the  distance  is  332  miles.  One-half  a  mile 
more  than  -|  of  this  distance  was  opened  in  Novem- 
ber, 1855,  and  the  remainder  in  November,  1856. 
Find  the  number  of  miles  opened  in  1856. 

12.  The  36  Israelites  who  fell  in  the  first  assault  on  Ai, 

were  y|^-  of  the  force  sent  by  Joshua.  How  many 
were  sent  by  Joshua? 

13.  What  number  multiplied  by  8|  equals  3£+£+f|+f-f  ? 

14.  Multiply  the  sum  of  -^j-  and  £  by  their  difference. 

15.  Of  the  distance  from   Edinburgh  to  London  by  rail, 

that  from  Edinburgh  to  Carlisle  is  J,  from  Carlisle 
to  Preston  -fo,  while  that  from  Preston  to  London  is 
210  miles.  Find  the  distance  from  Edinburgh  to 
London. 

16.  How  many  times  can  a  measure  holding  -J  of  a  pint  be 

filled  from  a  vessel  containing  63i  pints? 

17.  Of  a  consignment  of  guano  |-g-J  consisted  of  carbonate 

of  lime  and  phosphates  of  lime  and  magnesia,  and  the 
phosphates  made  up  ^-|  of  the  guano.  How  many 
parts  in  a  hundred  of  the  guano  was  carbonate  of 
lime? 

18.  Of  the  water  of  the  Dead  Sea  yff^  is  muriate  of  lime, 

-£fo  muriate  of  magnesia,  -fffo  muriate  of  soda,  2  fa  g 
sulphate  of  lime.  What  part  of  the  whole  do  these 
ingredients  constitute? 


142  COMMON   FRACTIONS. 


Ex.  96. 
ORAL  EXERCISE  IN  FRACTIONS. 

1.  From  a  piece  of  cloth  J  of  it  and  i  of  it  have  been 

cut.     What  fraction  of  the  cloth  is  left  ? 

2.  To  make  a  yard  of  cloth,  what  fraction  of  a  yard  must 

be  added  to  the  sum  of  J  and  J  of  a  yard  ? 

3.  A  boy  gave  to  his  sister  J  of  an  apple,  to  his  brother 

i  as  much  as  to  his  sister,  and  kept  the  rest  himself. 
What  part  of  the  apple  did  he  keep  ? 

4.  A  grocer  sold  }  of  a  dozen  eggs,  and  carried  home  the 

rest  of  the  dozen.     How  many  did  he  carry  home? 

5.  What  is  meant  by  J,  i,  J  of  a  unit? 

6.  How  is  J,  J,  J  of  a  unit  found? 

7.  How  is  J,  J,  i  of  a  number  found? 

8.  At  ?  of  a  dollar  per  yard,  what  is  the  cost  of  6  yards 

of  cloth  ? 

9.  At  $7  per  ton,  what  is  the  cost  of  i  of  a  ton  of  coal? 

10.  Three  packages  of  sugar  weigh  respectively  2i,  3},  4J 

pounds.     What  is  the  weight  of  the  whole  ? 

11.  When  poultry  is  worth  20  cents  per  pound,  what  must 

be  paid  for  a  turkey  weighing  8J  pounds,  and  a 
chicken  weighing  3i  pounds? 

12.  From  a  jar  of  butter  containing  15J  pounds  there  have 

been  sold  7}  pounds.     How  many  pounds  remain 
in  the  jar? 

13.  Change  to  mixed  numbers  Jf,  y,  if-,  *f-t  *£,  ff. 


COMMON   FRACTIONS.  143 

14.  Express  in  lowest  terms  |f ,  T\,  £$,  ff ,  ff ,  -nrfr- 

15.  Change  to  improper  fractions  5^-,  6-|,  8-|,  9^-,  13^. 

16.  Reduce  f  to  lOths  ;  to  15ths  ;  to  20ths ;  to  25ths. 

17.  A  lady  gave  %  a  dollar  to  her  daughter,  and  ^  of  a 

dollar  to  her  son.  What  fraction  of  a  dollar  did 
the  daughter  receive  more  than  the  son  ? 

18.  At  |-  of  a  dollar  per  bushel,  how  many  bushels  of 

apples  can  be  bought  for  $3? 

19.  Four  pecks  make  a  bushel.     If  21  pecks  be  sold  from 

a  bushel  of  cranberries,  how  many  pecks  remain  ? 

20.  A  gentleman  bought  2  pairs  of  gloves  at  $1J  a  pair, 

and  3  pairs  of  slippers  at  $1J  per  pair.  He  gave  a 
ten -dollar  bill  in  payment.  What  change  should 
he  receive? 

21.  What  part  of  2  is  1?  of  7  is  3?  of  9  is  2?  of  12  is  4? 

22.  A  farmer  planted  3  bushels  of  potatoes,  and  harvested 

50  bushels.  What  fraction  of  the  crop  was  the 
seed? 

23-    From  a  piece  of  cloth  containing  81  yards  there  were 
sold  45  yards.     What  part  of  the  piece  was  sold  ? 

24.  What  part  of  f  is  f  ?  of  |  is  £?  of  |  is  |? 

HINT.    Reduce  the  fractions  to  similar  fractions. 

25.  What  part  of  8  is  |  ?  of  7  is  -f  ? 

26.  In  a  year  there  are  365  days.     What  part  of  a  year 

are  30  days?  50  days?  75  days?  105  days? 

27.  Three-fourths  of  a  cord  of  oak  wood  costs  $6.     What 

is  the  cost  of     of  a  cord  ?  of  a  cord  ? 


144  COMMON   FRACTIONS. 

28.  Seven-eights  of  a  yard  of  cloth  cost  42  cents.     Find 

the  cost  of  i  of  a  yard ;  of  1  yard  ;  of  2i  yards. 

29.  Four-fifths  of  a  load  of  wood  is  sold  for  $8.     Required 

the  cost  of  £  of  the  load ;  of  the  whole  load ;  of  4f 
such  loads. 

;0.    What  is  the  price  of  a  bushel  of  turnips,  when  }  of  a 
bushel  are  sold  for  30  cents  ? 

31.  A  farmer  divided  among  his  4  sons  f  of  his  farm. 

What  part  of  the  farm  did  each  son  receive  ? 

32.  At  the  rate  of  $10  per  week,  what  is  the  cost  of  board 

per  day? 

33.  How  many  bushels  of  carrots,  at  $5  per  bushel,  can 

be  bought  for  $3i? 

34.  How  many  cows  are  £  of  20  cows?     16  sheep  are  £ 

of  how  many  sheep  ? 

35.  Five-sixths  of  12  hens  are  -|  of  how  many  hens? 

36.  Three-fourths  of  a  cord  of  wood,  at  $7  per  cord,  will 

pay  for  what  part  of  a  ton  of  coal,  at  $9  per  ton  ? 

37.  The  captain  of  a  vessel  owns  }  of  it,  the  first  mate  I, 

and  the  captain's  wife  \  of  the  remainder.     What 
part  of  the  vessel  does  she  own  ? 

38.  John  Rogers  sold  to  Henry  Cook  -J-  of  his  woodland, 

and  then  bought  back  J  of  what  he  had  sold.    What 
part  of  the  land  did  each  have  then  ? 

39.  From   a  bin  of  potatoes   containing   30  bushels,   5£ 

bushels,  2£  bushels,   4f  bushels  were   sold.     How 
many  bushels  were  left  in  the  bin  ? 


COMMON    FRACTIONS.  145 


40  A  bushel  of  wheat  weighs  60  pounds.  If  a  miller 
takes  3  pounds  from  each  bushel  for  toll,  what  part 
of  a  bushel  does  he  take  ? 

41.  At  20  cents  per  yard,  how  many  yards  of  ribbon  can 

be  bought  for  $2.20? 

42.  Four-fifths  of  $20  is  -§•  of  how  much  money? 

43.  Two-thirds  of  a  yard  of  silk  can  be  bought  for  $|. 

What  is  the  price  per  yard  ?     How  many  yards  can 
be  bought  for  $31? 

44.  If  5  bushels  of  oats  cost  $2,  what  will  be  the  cost  of 

9  bushels  at  the  same  rate  ? 

45.  If  $lj  are  paid  for  f  of  a  yard  of  velvet,  what  will  be 

the  cost  of  -|  of  a  yard  ? 

46.  If  -|  of  the  distance  between  two  towns  is  6|  miles, 

what  is  the  whole  distance  ? 

47.  If  2J   bushels  of  apples  make   a  barrel,   how   many 

barrels  will  11  bushels  make? 

48.  A  carpet  dealer  sold  £  of  £  of  a  roll  of  carpet.     What 

part  of  the  roll  was  left  ? 

49.  How    many    pigs    can   be   bought   for   $20,    at   $2J 

each? 

50.  Four  quarts  make  a  gallon.     When  2J  quarts  have 

been  taken  from  a  gallon  of  vinegar,  what  part  of 
the  gallon  has  been  taken  ? 

51.  A  drover  puts  -£•  of  his  cattle  in  a  field,  f  of  them  in 

another  field,  and  10  in  a  barn.     How  many  cattle 
has  he  ? 


146  COMMON   FRACTIONS. 

52.  A  merchant  sold  ^  of  a  chest  of  tea,  then  £  of  it,  and 

took  the  rest  home.  If  he  took  home  12  pounds, 
how  many  pounds  were  there  in  the  chest  at  first  r 

53.  A  cistern  has  two  pipes.     By  one  pipe,  3  gallons  of 

water  run  into  the  cistern  in  a  minute,  and  by  the 
other,  5  gallons  run  out  in  a  minute.  If  the  cistern 
contains  42  gallons,  and  both  pipes  are  open,  in  how 
many  minutes  will  it  be  emptied  ? 

54.  A  man  can  perform  a  certain  piece  of  work  in  4  hours, 

and  a  boy  can  do  the  same  work  in  6  hours.  What 
part  of  the  work  can  the  man  do  in  1  hour  ?  What 
part  can  the  boy  do  ?  What  part  can  both  together 
do?  How  many  hours  will  it  take  both  together 
to  do  the  work  ? 

65.  C  can  plant  an  acre  of  corn  in  6  hours,  C  and  D 
together  in  4  hours.  What  part  of  an  acre  can 
C  and  D  together  plant  in  1  hour  ?  What  part  can 
0  plant  in  1  hour  ?  What  part  then  can  D  plant 
in  1  hour  ?  How  many  hours  will  it  take  D  to 
plant  the  acre  of  corn  ? 

56.  A  fox  is  90  rods  in  advance  of  a  greyhound.     The 

fox  runs  60  rods  a  minute,  the  greyhound  65.  In 
how  many  minutes  will  the  fox  be  overtaken  ? 

57.  By  selling  cigars  at  $7  a  hundred  -fa  of  their  cost  is 

gained.  Find  the  price  per  hundred  at  which  they 
must  be  sold,  in  order  to  gain  f  of  their  cost. 

58.  By  selling  a  farm  for  $2400,  the  owner  lost  £  of  what 

it  cost  him.     How  much  did  he  pay  for  the  farm  ? 

59.  How  many  flowers  can  be  planted  along  the  borders 

of  a  flower-bed  12  feet  long  and  10  feet  wide,  if  the 
flowers  are  £  of  a  foot  apart  ? 


COMMON   FRACTIONS.  147 

Ex.  97. 

1.  Reduce  to  simple  fractions,  81  of  I  of  1,  -     — ,  -     ;-|. 

f   01  7     5  OI  f 

2.  Find  the  values  of 

169  —  14$;  !T9T--|-of4;  76}  —  f  of  19. 

3.  Six  pieces  of  cloth   measure  respectively  23^-  yards, 

19|  yards,  21£  yards,  24£  yards,  35f£  yards,  18f 
yards.  After  39^  yards  have  been  sold  from  their 
sum,  how  much  remains? 

4.  The  remainder  being   4,    the   quotient   51,    and   the 

divisor  25,  it  is  required  to  find  the  dividend. 

5.  Find  the  value  of  f  of  a  chest  of  tea  weighing  57  i 

pounds,  at  $1J  per  pound. 

6.  If  a  man  work  82  hours  in  a  day  he  can  finish  a  piece 

of  work  in  12 2  days.  How  many  hours  per  day 
must  he  work  to  complete  it  in  10 1  days? 

7.  A  confectioner  sells  f  of  %  of  a  bushel  of  walnuts. 

What  part  of  the  bushel  remains,  and  what  will  it 
bring  at  15  cents  per  quart? 

8.  What  is  the  value  of  a  basket  of  588  eggs,  worth  25 

cents  per  dozen  ? 

9  A  man  starts  on  a  journey  5  hours  before  the  mail 
coach.  How  many  miles  will  the  coach  be  ahead 
of  the  man  after  it  has  run  for  12  hours,  supposing 
that  he  travels  at  the  rate  of  3  miles  an  hour,  and 
the  coach  10  miles  an  hour? 

10.    If  f  of  |  of  a  piece  of  land  cost  $420,  what  is  the 
value  of  the  whole  ? 


148  COMMON   FRACTIONS. 

11.  A  farmer  sold  at  market  15  sheep  at  $2|-  each,  and 

bought  7  yards  of  cloth  at  $1£  per  yard.  How 
much  money  did  he  take  home  ? 

12.  A  man  walked  a  distance  of  60  miles  ;  for  the  first  5 

hours,  at  the  rate  of  3  miles  an  hour,  and  during 
the  remainder  of  the  journey  he  walked  at  the  rate 
of  4  miles  an  hour.  In  how  many  hours  did  he 
complete  the  journey  ? 

13.  The  circumference  of  a  fore  wheel  of  a  wagon  is  6-| 

feet ;  that  of  the  hind  wheel  8f  feet.  In  a  distance 
of  20  miles,  of  5280  feet  each,  how  many  more  turns 
will  be  made  by  the  former  than  by  the  latter  ? 

14.  A  young  man  received  $1200  from  his  father.     He 

spent  -J-  of  the  money  for  clothes,  %  of  it  in  travelling, 
and  invested  the  remainder  in  a  mortgage.  What 
fraction  of  the  whole  was  the  sum  invested  ? 

15.  A  baker  paid  $32  for  ^  of  a  hogshead  of  molasses. 

What  was  the  value  of  -J-  of  the  remainder  ? 

16.  A  gentleman  paid  $125  for  keeping  2  horses  12  weeks. 

What  would  it  cost,  at  the  same  rate,  to  keep  one 
horse  -£  of  a  week  ? 

o 

17.  If  -|  of  a  yard  of  ribbon  cost  $  -J,  what  will  be  the  value 

of  5f  yards? 

18.  Reduce  f,    £,    -f%  to  decimal  fractions,  and  add  the 

results. 

19.  The  contents  of  a  chest  of  tea  weighing  87.5  pounds 

are  made  up  into  1  pound,  ^  pound,  ^  pound  pack- 
ages, an  equal  number  of  each,  How  many  pack- 
ages of  each  kind  ? 


COMMON   FRACTIONS.  149 

20.  In  five  successive  days  a  farmer  puts  into  his  bin  371 

bushels  of  potatoes,  and  on  each  of  these  days  he 
sells  19f  bushels.  How  many  bushels  have  been 
put  into  the  bin  ?  How  many  more  are  in  the  bin 
at  the  end  than  at  the  beginning  of  the  five  days? 

21.  A    man's   weekly    income   is   $18i,  and  his  weekly 

expenses  are  $23?.  If  he  have  $75}  in  reserve, 
how  many  weeks  can  he  live  without  incurring 
debt? 

22.  By  a  leak  87-f  barrels  of  water  enter  the  hold  of  a 

boat  in  1  hour;  the  pumps  will  discharge  58f 
barrels  in  an  hour.  If  she  can  carry  only  875 
barrels,  in  how  many  hours  will  she  sink? 

23.  A  can  mow  a  field  in  10  days,  B  in  8  days,  and  C  in 

5  days.  When  wonting  together,  how  many  days 
will  they  need? 

24.  A  carpenter  alone  can  build  a  shop  in  15  days,  and 

with  the  help  of  his  son  he  can  build  it  in  10  days.' 
In  how  many  days  will  the  son  alone  build  the 
shop? 

25.  Wales  Edwards  and  George  Peters  hire  a  pasture  for 

$14.  Edwards  puts  in  8  horses  ;  Peters  puts  in  50 
sheep.  If  21  sheep  will  eat  as  much  as  2  horses, 
what  must  each  pay  ? 

26.  A  flour  dealer  bought  125  barrels  of  flour  at  $61. 

He  sold  97  barrels  at  $7f,  and  the  remainder,  being 
injured,  brought  only  $5£.  What  did  he  gain? 

27.  A  lady  bought  -|  of  -|  of  a  yard  of  ribbon  for 

What  was  the  cost  per  yard? 


150  COMMON   FRACTIONS. 

28.  From  two  fields  482  bushels  of  corn  are  gathered. 

The  first  field  yields  $  as  much  as  the  second.  How 
many  bushels  does  each  field  yield  ? 

29.  A  farmer  brought  to  market  3  jars  of  butter,  weighing 

26  pounds,  37  pounds,  19  pounds.  The  empty  jars 
weighed  3^  pounds,  4J-  pounds,  5|  pounds.  The 
butter  brought  $30.  What  was  the  price  per  pound  ? 

30.  From  120  acres  of  land  32£  acres  are  sold  to  one  man, 

and  -J-  of  the  remainder  to  another.  How  many 
acres  are  unsold  ? 

31.  If  the  rent  of  3  acres  of  land  for  -|  of  a  year  be  $9, 

what  will  be  the  rent  of  45  acres  for  1  year  ? 

32.  If  -|  of  a  ton  of  coal  cost  $4,  how  many  tons  can  be 

bought  for  $145£? 

33.  If  12  horses  eat  653  bushels  of  oats  in  3  months,  how 

many  bushels  will  7  horses  eat  in  2  years  ? 

34.  The  agent  for  a  line  of  steamers  sells  ^  of  a  steamship 

to  one  company,  J  of  the  remainder  to  a  second, 
and  •£  of  what  is  left  to  a  third.  What  part  of  the 
whole  ship  has  the  third  company  ? 

35.  A  farmer  exchanged  13  loads  of  oats,  of  18  bags  each, 

every  bag  containing  2J  bushels,  for  150  sheep,  at 
$2.925.  What  was  the  price  of  the  oats  per  bushel  ? 

36.  Two  men  95.784  miles  apart  approached  each  other 

until  they  met.  One  travelled  7.476  miles  more 
than  the  other.  How  many  miles  did  each  travel  ? 

37.  A  teacher  spent  -§•  of  his  salary  in  board  for  himself 

and  family,  and  ^  of  it  in  clothing  for  himself. 
The  clothing  of  his  wife  and  child  cost  f  as  much  as 
his  own.  At  the  end  of  the  year  $187  remained, 
What  was  the  salary  ? 


COMMON   FRACTIONS.  151 

38.  A  road  to  the  top  of  a  hill  has  a  rise  of  -^  of  a  foot  in 

100  feet.  How  many  feet  is  the  total  elevation  of 
the  hill,  if  the  length  of  the  road  is  2  miles? 

39.  A  man  bequeathes  to  his  wife  -J-  of  his  estate  ;  to  his 

daughter,  £  of  it;  to  his  son,  ^  of  the  daughter's 
share  ;  he  divides  the  remainder  equally  between  a 
hospital  and  a  public  library.  What  part  is  received 
by  the  hospital  ? 

40.  If  the  above  estate  is  worth   $150,784,  T^hat  is  the 

amount  received  by  the  hospital  ? 

41.  A  can  build  a  wall  in  7  days,  B  in  6  days,  and  C  in 

5  days.  A  and  B  worked  together  for  2  days, 
when  they  were  joined  by  C.  How  many  days 
will  they  need  to  complete  the  remainder  of  the 
work? 

42.  Find  the  cost  of  75,849  bricks,  at  $9.75  per  M. 

43.  A  lumberman  exchanged  50,495  feet  of  round  timber, 

at  $4£  per  M,  for  pork,  at  $20f  per  barrel.  How 
many  barrels  of  pork  did  he  receive  ? 

44.  For  %  of  a  bushel  of  apples  $|^  are  paid.     What  will 

4f  bushels  be  worth  ? 

45.  Henry  Jones  bought  at  a  saw-mill  3485  ft.  boards,  at 

$7.50  per  M  ;  9872  feet  laths,  at  $0.25  £er  C  ;  6492 
feet  flooring,  at  $85  per  M;  8975  feet  cherry 
boards,  at  $15.05  per  M.  He  paid  $152.75  in 
cash,  and  the  balance  in  flour,  at  $9.25  per  barrel. 
Required  the  number  of  barrels  of  flour. 

46.  A  merchant  mixed  7  pounds  of  black  tea  at  68  cents 

with  9  pounds  of  green  tea  at  75  cents.  At  what 
price  per  pound  must  he  sell  the  mixture  to  gain 
$3.69? 


152  COMMON   FRACTIONS. 


47.  Nine  men  working  10  hours  per  day  will  harvest  a 
piece  of  grain  in  8  days.  How  many  days  will  be 
needed  for  the  same  work  by  G  men  working  9  hours 
per  day  ? 


48. 


rv*  — j  ' 

At  $8.75  per  M,  how  many  bricks  can  be  bought  for 
$393.75? 

49.  When  1000  bricks  cost  $7.20,  what  is  the  cost  of  a 

single  brick  ? 

50.  If  $437.645  be  paid  for  6500  feet  of  rosewood,  what  is 

the  cost  per  M  ? 

51.  A  sea  captain  who  owned  $  of  a  ship  and  cargo,  gave 

to  his  wife  -^  of  his  share,  to  his  daughter  -J-  of  what 
his  wife  received,  to  his  son  -|  of  the  remainder,  and 
equally  divided  what  was  still  left  between  two 
nieces.  What  part  of  the  whole  had  each  niece  ? 

52.  Peter  Knowlton  sold  a  farm  for  $9786,  which  was  | 

of  the  sum  paid  for  it.  Required  the  original  cost 
of  the  farm. 

53.  A  merchant   bought  a  bag  of  coffee,   containing  60 

pounds,  for  $15.  At  what  advance  per  pound  must 
he  sell  it  to  buy  with  the  gain  on  the  coffee  3  yards 
of  velvet  at  $3  per  yard? 

54.  After  selling  -|  of  his  sheep  to  a  drover,  and  -J-  of  the 

remainder  to  his  neighbor,  a  farmer  has  150  left. 
How  many  were  there  in  the  flock  at  first? 

55.  A  stock  broker  bought  9  shares  in  the  Northern  Pacific 

Railroad,  at  $99},  and  12  shares  in  the  Illinois  Cen- 
tral Railroad,  at  $  102}.  He  sold  them  all  at  $  103  i 
How  much  did  he  gain  ? 


COMMON   FRACTIONS.  153 

56.  A    bankrupt's    available    property    can    be    sold    for 

$19,780,  which  will  pay  62J  cents  on  every  dollar 
he  owes.  How  much  does  he  owe  ? 

57.  A  loaf  of  bread  weighing  2  pounds,  when  flour  is  worth 

$  9.80  per  barrel,  is  sold  for  10  cents.  What  should 
it  bring  when  flour  is  worth  $7.84? 

58     Divide  0.75  of  17f  by  f  of  0.035. 

59.  An  army  of  7844  men  has  490,250  pounds  of  beef.     If 

for  every  man  11  pounds  daily  be  allowed,  in  how 
many  days  will  the  beef  be  consumed  ? 

60.  A  seedsman  bought  373  bushels  of  lawn  grass-seed  for 

$226.  He  sold  25  bushels  at  a  profit  of  $lf  per 
bushel.  For  what  price  per  bushel  must  he  sell  the 
remainder  to  make  his  whole  gam  $73? 

61.  The  cost  of  50  gallons  of  molasses  is  $25.     By  leakage 

-J-  of  it  is  lost ;  20  gallons  are  sold  at  62  J  cents.  At 
what  rate  must  the  remainder  be  sold  to  gain  $5  on 
the  whole? 

62.  For  |  of  a  yard  of  broadcloth  at  $6-}  per  yard,  1 J  yards 

of  cassimere  and  50  cents  in  money  were  given  in 
exchange.  What  was  the  price  per  yard  of  the 
cassimere  ? 

63.  A  owns  f  of  a  ship  and  cargo  worth  $25,748,  B  £  of 

the  remainder,  C  £  of  the  amount  belonging  to  A 
and  B,  and  D  owns  what  is  still  left.  Eequired  the 
amount  of  D's  share  ? 

64 o  A  farmer  gives  to  his  eldest  son  ^-|  of  a  farm,  and  the 
remainder  to  his  daughter.  The  difference  between 
their  shares  is  780  acres.  How  many  acres  does  the 
daughter  receive? 


154  COMMON   FRACTIONS. 

65.  If  1200  pounds  can  be  carried  36  miles  for  $14,  bow 

many  pounds  can  be  carried  24  miles  for  $  14  ? 

66.  If  2J-  acres  of  land  cost  $500,  wbat  will  460  acres  cost? 

87.  Four  and  four-sevenths  tons  of  cannel  coal  cost  $64. 
Required  tbe  cost  of  13f  tons. 

68  Of  a  certain  estate  £  is  pasture,  -|  land  suitable  for 
cultivation,  and  the  remainder,  woodland,  is  50 
acres.  How  many  acres  in  the  estate  ? 

69.  If  1.4  bushels  of  walnuts  cost  $1.50,  find  the  value  of 

7  bushels. 

70.  If  a  man  breathes  17  times  a  minute,  and  takes  in  at 

each  breath  -f-  of  a  quart  of  air,  how  many  quarts  of 
air  does  he  need  in  1  hour? 

71.  If  the  crop  of  potatoes  from  an  acre  is  on  the  average 

255  bushels,  but  the  potato  beetle  destroys  -|  of  the 
crop,  how  many  bushels  will  3£  acres  produce  ? 

72.  If  a  miller  takes  -^  for  toll,  and  a  bushel  of  wheat  pro- 

duces 40  pounds  of  flour,  how  many  bushels  must  be 
carried  to  the  mill  to  obtain  196  pounds  of  flour  ? 

73.  An  expressman  carried  100  vases,  on   the   condition 

that  he  was  to  receive  £  of  a  dollar  for  every  one  he 
carried  without  breaking,  and  pay  1J  dollars  for 
every  one  he  broke.  He  received  Ib  dollars.  How 
many  did  he  break? 

HINT.     The  expressman  loses  $9  on  the  lot,  and  he  loses 
$1J  on  each  vase  broken. 

74.  A  man  who  rows  4  miles  an  hour  in  still  water  takes  1^ 

hours  to  row  4  miles  up  a  river.    How  many  minutes 
will  it  take  him  to  row  4  miles  down  the  river  ? 
HINT.     The  man  rows  4f  miles  in  1J  hours.    Hence  the  cur- 
rent sets  him  back  f  of  a  mile  in  1  £  =  f  hours,  or  f  -*-  f  =  f  of 
a  mile  in  I  hour      In  rowing  down  the  river  he  rows  4  miles 
an  hour,  and  the  current  carries  him  f  of  a  mile  an  hour 


CHAPTER   IX. 

COMPOUND  QUANTITIES. 

142,  A  quantity  expressed  with  reference  to  a  single  unit 
is  called  a  simple  quantity;  but  a  quantity  expressed  with 
reference  to  different  units  is  called  a  compound  quantity, 

Thus,  201  pounds  is  a  simple  quantity,  but  20  pounds  4 
ounces  is  a  compound  quantity. 

143,  The  process  of  changing  the  unit  in  which  a  quan- 
tity is  expressed,  without  changing  the  value  of  the  quantity, 
is  called  reduction, 

144,  If  the  change  be  from  a  higher  denomination  to  a 
lower,  it  is  called  reduction  descending ;  if  from  a  lower  to 
a  higher,  it  is  called  reduction  ascending, 

Thus,  1  yard  =  36  inches  is  an  example  of  reduction 
descending ;  and  24  inches  =  2  feet  is  an  example  of  reduc- 
tion ascending. 

UNITS  OF  LENGTH. 

145,  12  inches  (in.)  =  1  foot  (ft.) 

3  feet  =  1  yard  (yd.). 

5}  yards,  or  16J  feet,  =  1  rod  (rd.). 

320  rods,  1760  yards,  or  5280  feet,  =  1  mile  (mi.). 

NOTE.  A  line  =  ^  in. ;  a  barleycorn  =  J  in.  ;  a  hand  (used  in 
measuring  the  height  of  horses)  =  4  in. ;  a  palm  =  3  in. ;  a  span  = 
9  in. ;  a  cubit  =  18  in. ;  a  military  pace  =  2J  ft. ;  a  chain  =  4  rds. ; 
a  link  =  TJ7  chain  ;  a  furlong  =  J  mi. ;  a  knot  (used  in  navigation) 
=  6086  ft. ;  a  nautical  league  =  3  knots  ;  a  fathom  (used  in  measuring 
depths  at  sea)  =  6  ft. ;  a  cable  length  =  120  fathoms. 


156  COMPOUND   QUANTITIES. 

Ex.   98.     (Oral.) 

1.  How  many  inches  in  1  yd.  ?  in  J  yd.  ?  in  J  yd.  ? 

2.  How  many  yards  in  180  in.  ?  in  48  in.  ?  in  45  in.  ? 

3.  How  many  yards  in  3  rds.  ?    in  4  rds.  ?    in  5  rds.  ? 

4.  How  many  feet  in  2  yds.  ?  in  2  rds.  ?  in  2  rds.  2  yds.? 

5.  How  many  rods  in  33  ft.  ?    How  many  yards  in  33  ft.  ? 

6.  In  \  mi.  how  many  rods  ?  yards  ?  feet  ? 

7.  How  many  rods  in  0.4  of  a  mile?  in  0.3?  in  0.7  ? 

8.  What  part  of  a  mile  are  160  rds.  ?  80  rds.  ?  40  rds.  ? 

9.  What  part  of  a  foot  are  4  in.  ?  3  in.  ?  6  in.  ?  8  in.  ? 
10.  What  part  of  a  yard  are  2  ft.  ?  1  ft.  6  in.  ?  2  ft.  6  in.  ? 

REDUCTION  DESCENDING. 
146,    Change  10  mi.  40  rds.  to  feet. 

10  mi.  40  rds. 
320 


3200 


40  10  X  320  rds.  =  3200  rds.,  to  which  the  40  rds. 


3240  are  added. 

Again,  3240  X  16J  ft.  =  53,460  ft. 


1620  The  multiplicand   and   multiplier  are  inter- 

19440  changed  in  the  operation. 

3240 

53460 

Ex.  99. 

Reduce  to  feet  :  Reduce  to  inches  : 

1.  3  mi.  5  yds.  2  ft.  4.    18  mi.  252  rds.  2  yds. 

2.  40  mi.  5  rds.  2}  yds.  5.    11  mi.  6  rds.  4  yds. 

3.  2  mi.  52  rds.  1  ft.  6.    18  mi.  230  rds.  8  ft. 


COMPOUND   QUANTITIES.  157 

7.  2  yds.  1  ft.  9  in.  10.    8  mi.  96  rds.  4  yds. 

8.  5  yds.  2  ft.  7  in.  11.    2  mi.  80  rds.  2  ft. 

9.  170  rds.  3  yds.  9  in.         12.    200  rds.  115  yds.  5  in. 

REDUCTION  ASCENDING. 

147,    Change  53,463  ft.  to  a  compound  quantity. 

16 }) 53463  ft. 

2 

33)  106926 half-feet. 

320)  3240  rds.    .  .  6  half-feet  =  3  ft. 
10  mi.    .  .  40  rds. 

10  mi.  40  rds.  3  ft.  Ans. 

There  are  16J  ft.,  qr  33  half-feet,  in  a  rod ;  so  the  53,463  ft.  are 
changed  to  half-feet,  and  the  half-feet  to  rods,  by  dividing  by  33. 
The  remainder  is  6  half-feet  =  3  ft. 

3240  rds.  are  changed  to  miles  by  dividing  by  320,  the  number  of 
rods  in  a  mile.  The  remainder  is  40  rds. 

Reduce  376,985  in.  to  higher  denominations. 


12 
3 
5J 

2 

376985  in. 

yds. 

31415  ft.  .  .  5  in. 

10471  yds.  .  2  ft. 

2 

11 
320 

20942  half-yards. 

1903  rds.    .  9  half-yards  =  4} 

5  mi.  .  .  303  rds. 

The  J  yd.  of  the  4J  yds.  should  be  reduced  to  lower  denominations, 
and  the  result,  1  ft.  6  in.,  added  to  the  2  ft.  5  in.     Thus, 

mi.     rds.      yds.    ft.     in. 

5     303     4     2     5 

1     6 

5     303     5     0  11 
5  mi.  303  rds.  5  yds.  0  ft.  11  in.  Ans. 


158  COMPOUND   QUANTITIES. 

Ex.  100. 
Reduce  to  higher  denominations  : 

1.  211  in.                6.    125,899m.  9.  348,164  in. 

2.  33,777  in.     6.  179,875  in.  10.  247,391  in. 

3.  142,737  in.    7.  87,476  ft.  11.  99,204  ft. 

4.  33,000ft.     8.  97,378yds.  12.  11,220ft. 

COMPOUND  ADDITION  AND  SUBTRACTION. 
148.  Add: 


ml. 

rds. 

yd.. 

n. 

In 

6 

120 

3 

2 

2 

18 

15 

1 

1 

6 

3 

215 

2 

2 

8 

28 

31 

2J 

0 

4 

£ 

=1 

6 

28      31     2      1  10 

28  mi.  31  rds.  2yds.  1  ft.  10  in.  Ans. 

Write  the  numbers  so  that  units  of  the  same  denomination  shall 
De  in  the  same  column.  The  sum  of  the  inches  is  16.  Divide  the 
16  in.  by  12  (12  in.  =  1  ft.).  The  result  is  1  ft.  4  in.  Write  4  under 
the  column  of  inches,  and  add  1  to  the  column  of  feet 

The  sum  of  the  feet,  including  the  1  ft.  from  the  16  in.,  is  6.  Divide 
by  3  (3  ft  =  1  yd.).  The  result  is  2  yds.  0  ft.  Write  0  under  the 
column  of  feet,  and  add  2  to  the  yards. 

The  sum  of  the  yards,  including  the  2  yds.  from  the  6  ft.,  is  8. 
Divide  by  5}  (5J  yds.  =  1  rd.).  The  result  is  1  rd.  2J  yds.  Write  2J 
under  the  column  of  yards,  and  add  1  to  the  rods. 

The  sum  of  the  rods,  including  the  1  rd.  from  the  8  yds.,  is  351. 
Divide  by  320  (320  rds.  =>  1  mi.).  The  result  is  1  mi.  31  rds.  Write 
31  under  the  column  of  rods,  and  add  1  to  the  miles. 

The  sum  of  the  miles,  including  the  1  mi.  from  the  351  rds.,  is  28. 

The  J  yd.  is  changed  to  1  ft,  6  in.  and  added  to  0  ft.  4  in. 


COMPOUND   QUANTITIES.  159 

149,    Take  4  mi.  110  rds.  5  yds.  2  ft.  from  6  mi.  25  rds. 
4  yds.  2  ft. 

mi.  rds.  yds.         ft. 

6        25       4       2 
4       110       5       1 


1       234      4£     1 

f=l     6  in. 

1       234       4       2     6 

1  mi.  234  rds.  2  ft.  6  in.  Ans. 

Write  the  numbers  so  that  units  of  the  same  denomination  shall 
be  in  the  same  column. 

Since  5  yds.  are  more  than  4  yds.,  1  rd.  is  taken  from  the  25  rds., 
and  reduced  to  yards,  and  the  result  added  to  4  yds.,  making  9J  yds. 
Then  9£  yds.  -  5  yds.  =  4J  yds. 

The  4J  is  written  under  the  column  of  yards. 

Since  110  rds.  are  more  than  24  rds.,  1  mi.  is  taken  from  the  6  mi., 
and  reduced  to  rods,  and  the  result  added  to  24  rds.,  making  344  rds. 
Then  344  rds.  -  110  rds.  =  234  rds. 

The  234  is  written  under  the  column  of  rods.  The  4  mi.  are  sub- 
tracted from  5  mi.  and  the  J  yd.  is  changed  to  1  ft.  6  in, 

Ex.  101 
Add: 

yds.        ft.         in.  rds.      yds.      ft.  mi.         rds.     yds 

1.    15  1  7  2.    23     3     1  3.    17  23  4 

23  2  9  18    4    2  9  17  2 

35  0  6  27    0    2  23  0  3 

7  2  11  640  11  35  1 


mi. 

rds. 

yds. 

mi. 

rds. 

ft. 

mi. 

rds. 

ft. 

in. 

4. 

37 

14 

2 

5.  23 

119 

15 

6.  7 

95 

8 

9 

28 

16 

2 

19 

173 

11 

8 

96 

7 

8 

19 

10 

4 

8 

65 

12 

3 

98 

9 

9 

10 

56 

3 

32 

147 

8 

6 

87 

8 

7 

160  COMPOUND   QUANTITIES. 

Find  the  difference  between  : 

yds.       ft.        In.  rds.      yds.     ft.  ml.          rds.        ft. 

7.    14     1       4          8.    22     2    0  9.   23       76     1 

10    2    11  19    3    2  6     157    2 


ml.     rds.       ft.     in.  mi.      rds.      yds.      ft.  ml.        rds.      yds. 

10.  17  125  1  10  11.  7   000  12.  13  33  2 
8  187  2  11     3  64  3  2      9  32  4 


COMPOUND  MULTIPLICATION  AND  DIVISION. 

150,  Multiply  37  yds.  2  ft.  11  in.  by  4. 

4  X  11  in.  =  44  in.  =  3  ft.  8  in.      Write  the 
yda         «,        in  8  in.  under  the  column  of  inches. 

37        2        ll  4  X  2  ft.  -  8  ft.;  8  ft  with  the  3  ft.  added 

4          are  11  ft.  =  3  yds.  2  ft.     Write  the  2  ft.  under 

151       2        8        ^e  c°lumn  °f  feet- 

4  X  37  yds.  =•  148  yds. ;  and  148  yds.  with 
the  3  yds.  added  =  151  yds. 

151  yds.  2  ft.  8  in.  Ans. 

NOTE.   When  the  multiplier  is  the  product  of  two  factors,  multiply 
by  one  of  the  factors,  and  the  resulting  product  by  the  other. 

151.  Divide  121  yds.  2  ft.  by  73. 

73)121     2  (lyd.  2ft. 
_73 

48 
3  The  remainder  from  dividing  121  yds.  by  73  is 

TT1  48  yds.,  which  are  reduced  to  feet  by  multiplying 

o  by  3  (3  ft.  =  1  yd.).     The  result  with  the  2  ft. 

r^  added  is  146  feet. 

-  .„  There  is  no  remainder  from  dividing  146  ft.  by 

iS2  73. 

1  yd.  2  ft.  Ans. 


COMPOUND    QUANTITIES.  161 

Divide  10  ft.  11  in.  by  2  ft.  8  in. 
Reduce  both  quantities  to  inches. 

10  ft,  11  in.  =  131  in.         jiai  =  4  8 
2  ft.    8  in.  =    32  in.  4^.  Ans. 

Ex.  102. 

1.  Multiply  33  yds.  2  ft.  11  in.  by  17. 

2.  Multiply  23  rds.  3  yds.  2  ft.  by  100. 

3.  Divide  15  yds.  1  ft.  9  in.  by  3. 

4.  Divide  289  yds.  2  ft.  9  in.  by  213. 

5.  Divide  150  mi.  178  rds.  3  yds.  by  9. 

6.  Multiply  3  mi.  72  rds.  9  ft.  by  11. 

7.  Multiply  150  rds.  2  yds.  1  ft.  by  235. 

8.  Divide  33  mi.  40  rds.  by  200. 

9.  Divide  200  mi.  56  rds.  3  yds.  2  ft.  by  121. 

10.  Multiply  11  mi.  200  rds.  by  14. 

11.  Multiply  52  mi.  1021  yds.  by  47. 

12.  Divide  43  mi.  280  rds.  by  24. 

FRACTIONS  OF  SIMPLE  AND  COMPOUND  QUANTITIES. 

152,    Express  -|  of  a  mile  in  rods,  feet,  and  inches. 
|  mi.  =  -§•  of  320  rds.  =  213£  rds. 
£rd.  ="|of  16^  ft.     -5^  ft. 
^  ft.  =$  of  12  in.      =  6  in. 

213  rds.  5  ft.  6  in.  Ans. 


162  COMPOUND   QUANTITIES. 

Express  0.6275  of  a  mile  in  rods,  feet,  and  inches. 

0.6275 
320 


12  5500  0.6275  mi.  =  0.6275  of  320  rds.  =  200.8  rds. 

18825  0.8  rd.        =  0.8  of  16J  ft.          =  13.2  ft. 

200.8  rds.  0.2  ft.         =0.2  of  12  in.  =  2.4  in. 


13.2  ft.  200  rds.  13  ft.  2.4  in.  Am. 

12 
2.4  in. 

Find  the  value  of  f  of  3  rds.  14  ft.  7  in. 

o      14        tj  Here  we  multiply  by  the  numerator  of  the 

c        fraction,  and  divide  the  product  by  the  denomi- 
nator. 


9)19      6     11 

227-|  2  rds.  2  ft.  7f  in.  Ans. 

NOTE.    When  the  multiplier  is  a  mixed  number,  multiply  by  the 
integer  and  the  fraction  separately,  and  add  the  resulting  products. 

Ex.  103. 
Find  the  value  of : 

1.  |  of  a  mile.  4.    ^  mi.  +  £  of  40  rds.  +  f  yd. 

2.  %  of  a  mile.  5.    0.475  of  a  mile. 

3.  |  mi.  —  -f  rd.  6.    0.3975  of  a  mile. 

7.  0.01284  of  14  miles. 

8.  3.726  mi.  -  33.57  rds. 
9.    Find  |  of  5  mi.  89  rds.  3  yds.  2  ft. 

10,  Take  £  of  4  mi.  from  f  of  3  mi.  18  rds.  3  yds.  2  ft. 

11.  Add  0.525  mi.,  0.125  rd,  0.5  yd,  0.16  ft. 


COMPOUND   QUANTITIES.  163 


To   EXPEESS   ONE  QUANTITY   AS  THE   FRACTION   OF  ANOTHER, 

153,    Express  145  rds.  2  yds.  1  ft.  6  in.  as  the  fraction 
of  a  mile. 

6in.  =6?ft-    =     &• 


2^  yds.       =       ras.=.&rcL 

Of 

1600 


^P  of  a  mile.  -4ns. 

Express  120  rds.  3  yds.  1  ft.  6.72  in.  as  the  decimal  of  a 

mile. 

6.72  in.  -i-  12  =  0.56  ft.,  and   this  added  to 
12 
3 

5.5 
320 


6.72  in.  the  1  ft.  gives  1.56  ft.     1.56  ft.  -i-  3  =  0.52  yds., 

1.56  ft.  and  this  added  to  3yds.  gives  3.52  yds.     3.52 

3.52  yds.  yds.  -i-  5.5  gives  0.64  rds.,  and  this  added  to 

120.64  rds.  120  rds.  gives  120.64  rds.      120.64  rds.  -*-  320 


0.377  mi.       gives  0.377  mi. 

0.377  of  a  mile.  Ans. 

NOTE.  The  quotient  in  any  case  need  not  be  carried  beyond  the 
fifth  decimal  place,  and  the  required  answer  will  be  sufficiently  accu- 
rate for  all  practical  purposes. 

J54,    Express  1  yd.  2  ft.  3  in.  as  the  fraction  of  5  yds. 
1  ft.  3  in. 
1  yd.  2  ft.  3  in. :  5  yds.  1  ft.  3  in. 

Sin.  =^  ft.    ==Jft.  Sin.  =  ^  ft.    =  £  ft. 

2i  ft-  =  f  yd.  -|  yd.  H  ft.  =  ii  yd.  =  &  yd. 


If  yds.  5-j^-  yds. 


ft.  Ans. 


164  COMPOUND   QUANTITIES. 

NOTE.  If  the  answer  to  the  last  problem  is  to  be  expressed  as  a 
decimal  fraction,  first  find  the  answer  as  a  common  fraction,  and 
reduce  this  common  fraction  to  a  decimal  fraction. 

Ex.  1O4. 
Express : 

1.  125  rds.  4  yds.  2  ft.  6  in.  as  the  fraction  of  a  mile. 

2.  1  yd.  2  ft.  3  in.  as  the  fraction  of  5  yds. 

3.  51  rds.  1  yd.  3.6  in.  as  the  decimal  of  a  mile. 

4.  %  rd.  +  £  yd.  as  the  fraction  of  a  mile. 

5.  o  mi.  53  rds.  4  yds.  1.2  ft.  as  the  decimal  of  5  mi. 

89  rds.  3  yds.  2  ft. 

6.  2  mi.  138  rds.  1  yd.  as  the  fraction  of  3  mi.  265  rds. 

3£  yds. 

7.  233  rds.  9  ft.  10.8  in.  as  the  decimal  of  a  mile. 

8.  3  mi.  242  rds.  2|  yds.  as  the  decimal  of  7  mi.  160  rds. 

9.  2  ft.  7|  in.  as  the  decimal  of  100  yds. 

10.  11  rds.  4  yds.  4^-  in.  as  the  fraction  of  a  mile. 

11.  -J  rd.  +  f  yd.  +  -fo  ft.  as  the  fraction  of  a  rod. 

12.  195  yds.  1  ft.  8  in.  as  the  fraction  of  -J-  of  a  mile. 

13.  1  mi.  232  rds.  4  yds.  1  ft.  6  in.  as  the  fraction  of  8  mi. 

204  rds.  0  yd.  1  ft.  6  in. 

14.  127  rds.  3  ft.  3.6  in.  as  the  decimal  of  a  mile. 

15.  261  rds.  4  yds.  1  ft.  6  in.  as  the  fraction  of  a  mile. 

16.  f  of  the  difference  between  3  yds.  2  ft,  11  in.  and  10 

yds.  7  in.  as  the  fraction  of  16  yds. 

17.  7  rds.  1  ft.  3.17  in.  as  the  decimal  of  76  rds.  2  yds.  5  in. 

18.  248  rds.  4  yds.  2  ft.  8  in.  as  the  fraction  of  2  mi. 


COMPOUND   QUANTITIES. 


165 


MEASURES  OF  SURFACE. 

155.  A  surface  has  two  dimensions,  length  and  breadth. 

156.  If  a  surface  is  flat  and  has  four  square  corners,  it  is 
called  a  rectangle, 

157.  If  a  rectangle  has  its  four  sides  equal,  it  is  called  a 
square. 


Rectangle, 


158.  The  unit  of  surface  is  a  square  each  side  of  which 
is  a  linear  unit. 

159.  The  area  of  a  surface  is  the  number  of  square  units 
it  contains. 

160.  Suppose  the  rectangle  in  the  margin  is  3  in.  long 

and  2  in.  wide.      If  lines  are  drawn 

as  represented  in  the  figure,  the  sur- 
face will  be  divided  into  square  inches. 

There  will  be  2  horizontal  rows  of  3 
square  inches  each;  that  is,  in  all, 
2x3  square  inches.  Hence, 

Express  the  length  and  breadth  of  a  rectangle  in  the  same 
linear  unit ;  the  product  of  these  two  numbers  will  express 
its  area  in  square  units  of  the  same  name  as  the  linear  unit 
of  the  sides. 

Conversely,  the  number  of  square  units  in  a  rectangle 
divided  by  the  number  of  linear  units  in  one  side  will  give 
the  number  of  linear  units  in  its  adjacent  side. 


166  COMPOUND   QUANTITIES. 


UNITS  OF  SURFACE. 

161,     144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.). 

9  square  feet  =  1  square  yard  (sq.  yd.). 

30}  square  yards,  or  j  =  l  d  (       rf } 

272}  square  feet, 

160  square  rods,  or  j  =  1  acre  , A  . 

10  square  chains,  ) 

640  acres  =  1  square  mile  (sq.  mi.). 


A  square  of  flooring  or  roofing  —  100  sq.  ft. 
A  section  of  land  =  1  mile  square. 

A  township  =  36  sq.  mi. 

The  units  of  surface  measure  are  obtained  by  squaring  the  units 
of  linear  measure.     Thus, 

144  =  12";  9-3';  30}  -(5J)2;  272}  =  (16J)2 


Ex.  105.    (Oral.) 

1.  How  many  square  feet  of  surface  in  a  blackboard  4  ft. 

wide  and  9  ft.  long  ? 

2.  If  a  slate  is  8  in.  wide  and  has  a  surface  of  80  sq.  in., 

what  is  the  length  of  the  slate  ? 

3.  How  many  square  inches  in  J  of  a  square  foot?    in  f  ? 

in  |?  in  |? 

4.  How  many  square  feet  in  3  sq.  yds.  ?  in  5  sq.  yds.  ? 

5.  How  many  square  inches  in  a  board  4  in.  long  and 

3  in.  wide  ? 

6.  A  square  yard  of  carpet  is  3  ft.  long  and  3  ft.  wide  ; 

how  many  square  feet  in  it  ? 


COMPOUND   QUANTITIES.  167 

7.  How  many  square  feet  in  a  yard  of  carpet  2  ft.  wide  ? 

2J  ft.  wide? 

8.  How  many  square  feet  in  a  room  12  ft.  by  15  ft.  ? 

9.  How  many  yards  of  carpet  2  ft.  wide  will  be  required 

to  cover  the  floor  of  the  above  room,  if  the  strips 
run  lengthwise  of  the  room  ? 

10.  How  many  square  yards  in  81  sq.  ft.  ? 

11.  How  many  square  rods  in  f  of  an  acre? 

12.  What  part  of  an  acre  are  40  sq.  rds.  ?  80?  100? 

Ex.  106. 

1.  Reduce  5  A.  147  sq.  rds.  to  square  rods. 

2.  How  many  square  inches  in  9  sq.  yds.  7  sq.  ft.  ? 

3.  Reduce  33,796  sq.  in.  to  square  yards. 

4.  Reduce  153  A.  87  sq.  rds.  to  square  inches. 

5.  In  67,413  sq.  yds.  how  many  acres? 

6.  In  a  rectangular  field  49  yds.  long  and  16  yds.  wide, 

how  many  square  feet  ? 

7.  How  many  tile.s  1  ft.  square  will  be  needed  to  pave  a 

hall  20  ft.  long  and  9  ft.  wide? 

8.  How  much  greater  is  the  area  of  a  lot  50  rds.  square 

than  that  of  a  lot  containing  50  sq.  rds.  ? 

9.  How  many  square  yards  in  a  square  lot  measuring 

142  ft.  on  a  side  ? 

10.    Ingrain  carpet  is  3  ft.  wide.     How  many  yards  will 
be  required  for  a  room  27  ft.  long  and  18  ft,  wide? 


168  COMPOUND   QUANTITIES. 


11.  From   each  corner  of  a  square,  the  side  of  which  is 

2  ft.  5  in.,  a  square  measuring  5  in.  on  a  side  is  cut 
out.     Find  the  area  of  the  remainder  of  the  figure. 

12.  Find  the  value  of  0.45  of  an  acre. 

13.  Reduce  ^J  of  a  square  mile  to  lower  denominations. 

14.  Reduce  80  sq.  rds.  2.42  sq.  yds.  to  the  decimal  of  an 

acre. 

15.  Add  £  of  an  acre,  $  sq.  rd.,  and  -|  sq.  yd. 

16.  Add  -|  of  an  acre  and  f  of  a  square  rod. 

17.  From  -j^-  of  a  square  rod  take  £  of  a  square  yard. 

18.  Find  -f  of  9  A.  70  sq.  rds.  15  sq.  yds.  7  sq.  ft.  19  sq.  in. 

19.  A  side  of  Russell  Square  in  London  is  660  ft.     How 

many  acres  does  it  contain  ? 

20.  The  area  of  a  rectangular  field  is  33  sq.  rds.  1  sq.  yd. 

6  sq.  ft.  108  sq.  in.,  and  the  length  is  9  rds.  1  ft. 
Gin.     What  is  the  width? 

The  area  of  a  circle'is  found  by  multiplying  the  square  of 
its  radius  by  3.1416.     (The  radius  is  half  the  diameter.) 

21.  Find  the  area  of  a  circular  pond  if  its  radius  is  300  ft. 

22.  Find  the  area  of  the  bottom  of  a  round  cistern  if  its 

diameter  is  11  ft. 

23.  The  radius  of  the  rotunda  of  the  Pantheon  of  Rome  is 

71  ft.  6  in.     Find  the  area  of  the  floor  in  square 
feet. 

24.  The  two  dials  of  the  clock  of  St.  Paul's,  London,  are 

each  18|  ft.  in  diameter.     Find  the  area  of  each 
in  square  feet, 


COMPOUND   QUANTITIES.  169 


CAEPETING  ROOMS. 

In  determining  the  number  of  yards  of  carpeting  required  for  a 
room,  we  first  decide  whether  the  strips  shall  run  lengthwise  or  across 
the  room,  and  then  find  the  number  of  strips  needed.  The  number 
of  yards  in  a  strip,  including  the  waste  in  matching  the  pattern,  mul- 
tiplied by  the  number  of  strips  will  give  the  required  number  of  yards. 

25.  How  many  yards  of  carpet  21  ft.  wide  will  cover  a 

floor  18  ft.  by  15  ft.,  if  the  strips  run  across  the  room  ? 

HINT.  18-*-  2J-  =  8.    Hence  8  strips  are  required.    15ft.  =  5  yds., 
and  8x5  yds.  =  40  yds. 

26.  How  many  yards  of  carpet  I  of  a  yard  wide  will  be 

required  for  a  floor  26  ft.  by  15J  ft.,  if  the  strips  run 
lengthwise?  If  the  strips  run  across  the  room? 
How  much  will  be  turned  under  in  each  case? 

27.  How  many  yards  of  carpeting  I  of  a  yard  wide  will  be 

required  for  a  room  8J  yds.  long  and  17  ft.  wide,  if 
the  strips  run  lengthwise  and  there  is  a  waste  of  -fa 
of  a  yard  in  each  strip  in  matching  patterns? 

28.  Find  the  cost  of  carpet  30  inches  wide,  at  $1.25  per 

yard,  for  a  room  18  ft.  by  14  ft.,  if  the  strips  run 
lengthwise.  If  the  strips  run  across  the  room. 

29.  Find  the  cost  of  carpeting  J  of  a  yard  wide,  at  $2.75 

per  yard,  for  a  room  34  ft.  8  in.  by  13  ft.  3  in.,  if 
the  strips  run  lengthwise,  and  if  there  is  a  waste  of 
i  of  a  yard  on  each  strip  in  matching  the  pattern. 

30.  Which  way  must  the  strips  of  carpet  J  of  a  yard  wide 

run  in  order  to  carpet  most  economically  a  room  20 
ft.  6  in.  long  and  19  ft.  6  in.  wide,  if  there  is  no 
waste  for  matching  the  pattern  ? 


170  COMPOUND   QUANTITIES. 


PAPERING  AND  PLASTERING. 

The  area  of  the  four  walls  of  a  room  is  equal  to  that  of  a  rectangle 
whose  length  is  equal  to  the  perimeter  of  the  room,  and  whose 
breadth  is  equal  to  the  height  of  the  room. 

31.  How  many  yards  of  plastering  in  the  four  walls  of  a 

room  14  ft.  3  in.  by  13J  ft.,  and  7  ft.  high,  if  no 
allowance  is  made  for  doors  and  windpws  ? 

HINT.     Perimeter  =  2  X  14J  +  2  x  13J  =  55J  ft. 

(7  X  55J)  sq.  ft.  =  386i  sq.  ft. ;  386i  -H  9  =  43  8q.  y<ls. 

32.  Find  the  yards  of  plastering  in  the  walls  of  a  room 

211  ft.  long,  16}  ft.  wide,  and  11  ft.  high,  if  12  sq. 
yds.  be  allowed  for  doors,  windows,  and  base- 
boards? 

33.  How  many  square  yards  of  plastering  in  the  walls  and 

ceiling  of  a  room  30  ft.  8  in.  long,  26  ft.  5  in.  wide, 
10  ft.  6  in.  high,  if  24  sq.  yds.  be  allowed  for  doors, 
windows,  and  base-boards? 

34.  What  will  be  the  cost  of  plastering  the  walls  and  ceil- 

ing of  a  room  27  ft.  4  in.  long,  20  ft.  wide,  and  12  ft. 
6  in.  high,  at  27  cents  per  square  yard,  if  20  sq.  yds. 
be  deducted  for  doors,  windows,  and  base-boards  ? 

35.  Find  the  cost  of  whitening  the  ceiling  and  walls  of  a 

room  14  ft.  4  in.  wide,  15  ft.  6  in.  long,  and  10  ft. 
6  in.  high,  at  5  cents  per  square  yard,  allowing  9 
sq.  yds.  for  doors  and  windows. 

36.  Find  the  cost  of  papering  a  room  32  ft.   long,  22  ft. 

wide,  13  ft.  high,  with  paper  18  in.  wide,  8  yds.  in 
a  roll,  at  $1.25  a  roll,  if  50  sq.  yds.  be  allowed  for 
doors,  windows,  and  base-boards. 


COMPOUND   QUANTITIES.  171 

i 

SURVEYORS'  CHAIN. 

162,    Surveyors  use,  in  measuring  distances,  a  chain  4 
rds.  long  and  containing  100  links. 

The  links  can  be  written  as  decimals  of  a  chain. 

Ex.  107.    (Oral.) 

1.  How  many  chains  make  a  mile? 

2.  In  10  mi.  how  many  chains? 

3.  How  many  rods  in  9  chains? 

4.  How  many  yards  in  1  chain?  how  many  feet? 

5.  How  many  chains  in  48  rods? 

6.  In  0.75  of  a  mile  how  many  chains? 

7.  How  many  chains  in  f  of  a  mile? 

8.  If  a  square  field  measures  1  chain  on  a  side,  how  many 

square   rods   does   it   contain  ?     How  many  square 
rods  then  in  a  square  chain? 

9.  If  a  square  chain  contains  16  sq.  rds.,  how  many  square 

chains  make  an  acre  ? 

10.  What  part  of  an  acre  are  4  sq.  chains?  5  sq.  chains? 

8  sq.  chains  ? 

11.  What  part  of  a  chain  are  50  links?  25  links?  20  links? 

12.  How  many  square  rods  in  a  square  garden-plot  meas- 

uring 75  links  on  a  side  ? 

13.  How  many  inches  long  is  a  link  ? 

14.  The  distance  between  two  places  is  found  to  be  320 

chains.     Express  the  distance  in  miles. 


172  COMPOUND   QUANTITIES. 

Ex.  108. 

1.  Redupe  38  chains  80  links  to  the  decimal  of  a  mile. 

2.  The  four  sides  of  a  field  are  23  chains  19  links,  17 

chains  34  links,  6  chains  85  links,  and  24  chains  G2 
links.     How  many  yards  around  the  field  ? 

3.  One  field  contains   3   sq.    chains,    and   another   is   3 

chains   square.     How   many   acres   in   both   fields 
together  ? 

4.  A  field  is  crossed  by  a  driveway  15  links  wide,  and 

13  chains  43  links  long.     How  many  square  rods  in 
the  driveway? 

5.  From  a  field  of  4  A.  a  rectangular  piece  3  chains  25 

links  long  and  2  chains  75  links  wide  is  reserved. 
How  much  of  the  field  is  left  ? 

6.  The  sides  of  a  triangular  field  measure  21  §  chains, 

14  chains  11  links,  and  8  chains  10  links  respec- 
tively.    By  how  many  yards  is  the  longest  side  less 
than  the  sum  of  the  other  two  ? 


BOARD  MEASURE. 

163,  Boards  one  inch  or  less  in  thickness  are  sold  by 
the  square  foot. 

Boards  more  than  one  inch  in  thickness,  and  all  squared 
lumber,  are  sold  by  the  number  of  square  feet  of  boards 
one  inch  in  thickness  to  which  they  are  equivalent. 

Thns,  a  board  16  ft.  long,  1  ft.  wide,  and  1  in.  thick,  contains  16 
ft.  board  measure.  If  only  J,  f ,  or  J  of  an  inch  thick,  it  still  contains 
16  ft. ;  but,  if  1 J  in.  thick,  it  contains  20  ft.  board  measure. 


COMPOUND   QUANTITIES. 


173 


Ex.  109. 
How  many  feet  board  measure  in  : 

1.  A  board  18  ft.  long,  6  in.  wide,  £  in.  thick? 

2.  A  board  16  ft.  long,  12  in.  wide,  1  in.  thick? 

3.  Forty  boards  14  ft.  long,  10  in.  wide,  J  in.  thick  ? 

4.  A  stick  of  square  timber  8  in.  by  9  in.,  and  30  ft.  longf 

5.  Six  joists,  each  3  in.  by  4  in.,  and  11  ft.  long? 

6.  Ten  joists,  each  6  in.  by  4  in.,  and  14  ft.  long? 

7.  A  stick  of  square  timber  10  in.  by  12  in.,  and  36  ft. 

long? 

8.  Ten  2-in.  planks,  each  13  ft.  long,  15  in.  wide? 

9.  Thirty  3-in.  planks,  each  12  ft.  long,  10  in.  wide? 

10.  A  board  24  ft.  long,  23  in.  wide  at  one  end,  and  17  in. 
at  the  other,  and  1£  in.  thick? 


HINT.   The  average  width  is 


23  +  1  7 


in. 


11.  In  a  stick  of  timber  40  ft.  long  and  15  in.  square? 

12.  Ten  4-in.  planks  16  ft.  long  and  10  in.  wide? 

MEASURES  OF  VOLUME. 

164,   A  rectangular  solid  is  a  solid  bounded  by  six  rect 
angles. 


165,   If  the  rectangles  are  all  squares,  the  solid  is  called 
a  cube, 


174 


COMPOUND   QUANTITIES. 


166,  In  the  figure  represented  in  the  margin,  let  the 
length  contain  5,  the  breadth  3,  and  the 
height  7  in. 

The  base  may  be   divided  into  square 
inches ;  there  will  be  three  rows  of  5  sq.  in. 
each ;  in  all  15  sq.  in.     Upon  each  square 
inch  may  be  placed  a  pile  of  7  cu.  in.,  so 
that  the  solid  will  contain  15  X  7  cu.  in. ; 
that  is,  3  X  5  X  7  cu.  in. 
Hence,  to  find  the  volume  of  a  rectangular  solid, 
Express  its  length,  breadth,  and  height  in  the  same  linear 
unit;  the  product  of  these  numbers  will  express  its  volume 
in  cubic  units  of  the  same  name  as  the  linear  unit. 


UNITS  OF  VOLUME. 

167.  1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (en.  ft.). 

27  cubic  feet  =  1  cubic  yard  (cu.  yd.). 

The  units  of  volume  are  cubes  of  the  linear  units.    Thus,  1 728  =  123, 
27  =  3s. 

168,  In  measuring  wood,  a  pile  8  ft.  long,  4  ft.  wide,  and 
4  ft.  high,  is  called  a  cord. 

X 


Hence, 


A  cord  foot  is  one  foot  in  length  of  such  a  pile. 


1  cord  foot  (cd.  ft.)  —  16  cu.  ft. 
8  cord  feet  »  1  cord  fed.), 


COMPOUND   QUANTITIES.  175 

Ex.110.      (Oral.) 

1.  A  brick  is  2  by  4  by  8  in.     Find  its  volume. 

2.  How  many  cubic  feet  in  2  cu.  yds.  ? 

3.  How  many  cubic  yards  in  81  cu.  ft.  ? 

4.  Twelve  cubic  feet  are  what  part  of  a  cubic  yard  ? 

5.  How  many  cords  in  40  cd.  ft.  ?  in  16  cd.  ft. 

6.  How  many  cords  in  a  pile  of  wood  32  ft.  long,  4  ft. 

wide,  and  4  ft.  high? 

7.  How  many  loads  of  earth,  each  1  cu.  yd.,  must  be 

removed  in  digging  a  ditch  21  ft.  long,  3  ft.  wide, 
and  3  ft.  deep  ? 

8.  How  many  cubic  feet  in  a  stick  of  timber  12  in.  wide, 

9  in.  thick,  and  24  ft.  long? 

9.  How  many  feet  board  measure  in  a  cubic  foot? 

10.    How  many  cubic  feet  in  a  stick  of  timber  16  in.  wide, 
9  in.  thick,  and  21  ft.  long? 

Ex.  111. 

1.  Reduce  15  cu.  yds.  13  cu.  ft.  to  cubic  inches. 

2.  Reduce  150,000  cu.  in.  to  cubic  yards. 

3.  Subtract  28  cu.  yds.  25  cu.  ft.  1500  cu.  in.  from  47 

cu.  yds.  13  cu.  ft.  1236  cu.  in. 

4.  Multiply  17  cu.  yds.  17  cu.  ft.  187  cu.  in.  by  11. 

5.  Divide  22  cu.  yds.  10  cu.  ft.  933  cu.  in.  by  7. 

6.  How  many  cubic  inches  can  be  cut  out  of  a  cubic  foot? 

7.  How  many  cubic  feet  of  water  will  a  cistern  hold  whose 

three  dimensions  are  each  4  ft.  ? 


176  COMPOUND   QUANTITIES. 

8.  How  many  cubic  inches  in  a  rectangular  stone  post 

3  ft.  high,  1  ft.  wide,  and  1  ft.  thick  ? 

9.  Find  the  value  of  0.975  of  a  cubic  yard. 

10.  Reduce  13  cu.  ft.  864  cu.  in.  to  the  decimal  of  a  cubic 

yard. 

11.  A  man  bought  52  cu.  yds.  18  cu.  ft.  984  cu.  in.  of  stone 

for  the  cellar  of  a  house,  and  |  as  much  for  the  cellar 
of  a  second  house.    How  much  did  he  buy  for  both  ? 

12.  How  many  cords  of  wood  in  a  pile  50  ft.  long,  6  ft. 

high,  and  4  ft.  wide  ? 

13.  How  many  cords  of  wood  in  a  pile  42  ft.  long,  6|-  ft. 

high,  and  8  ft.  wide  ? 

14.  What  must  be  the  length  of  a  load  of  wood  that  is 

4  ft.  wide,  5£  ft.  high,  to  contain  2  cds.  ? 

15.  A  cubic  foot  of  wood  weighs  20  pounds.     Find  the 

weight  of  10  boards,  each  30  ft.  long,  1  ft.  wide,  and 
1  in.  thick. 

UNITS  OF  CAPACITY. 
Dry  Measure. 

170,  2  pints  (pt.)  =  1  quart  (qt.). 
8  quarts        =  1  peck  (pk.). 

4  pecks         =  1  bushel  (bu.). 

Liquid  Measure. 

171,  4  gills  (gi.)    =  lpint(pt.). 
2  pints  =  1  quart  (qt.). 

4  quarts        =  1  gallon  (gal.). 


31J  gallons  =  1  barrel  (bbl.). 
2  barrels      =  1  hogshead  (hhd.). 

NOTE.  The  gallon  of  liquid  measure  contains  231  cu.  in.  The 
bushel  of  dry  measure  contains  2150.42  cu.  in.  Therefore  the  quart 
of  liquid  measure  contains  57f  cu.  in.,  and  the  quart  of  dry  measure 
67£  cu.  in. 


COMPOUND   QUANTITIES.  177 

Ex.   112.      (Oral.) 

1.  How  many  pints  in  16  gi.  ?  in  37  gi.  ? 

2.  How  many  pints  in  2  qts.  ?  in  7  qts.  1  pt.  ? 

3.  How  many  quarts  in  2  pks.  ?  in  3  pks.  3  qts.  ? 

4.  How  many  quarts  in  3  bu.  ? 

5.  How  many  baskets  holding  2^-  pks.  each  will  10  bu. 

of  apples  fill  ? 

6.  If  a  pint  of  milk  cost  4  cts.,  what  will  a  gallon  cost? 

7.  How  many  times  will  a  gallon  of  water  fill  a  half-pint 

cup? 

8.  How  many  pint  bottles  will  be  required  to  hold  5  gals. 

2  qts.  of  cider  ? 

9.  If  4  qts.  of  blueberries  cost  32  cts.,  what  will  a  bushel 

cost  at  the  same  rate  ? 

10.  A  2-gal.  measure  of  molasses  lacks  3  pts.  of  being  full. 

What  is  the  molasses  worth  at  80  cts.  a  gallon  ? 

11.  If  a  horse  eats  4  qts.  of  corn  a  day,  how  many  days 

will  a  bushel  last  him  ? 

12.  If  a  quart  of  berries  is  worth  10  cts.,  what  is  a  peck 

worth  ? 

Ex.  113. 

1.  Reduce  440  pts.  to  pecks,  and  109  pts.  to  gallons. 

2.  Reduce  2024  pts.  to  bushels. 
Add: 

gals.    qts.      pts.  bu.     pks.     qts.  bu.     pks.     qts. 

3.  13     2     1  4.    5     1     3  5.    17     2    4 
*       2     3     0                    611  11     34 

15     00  200  300 

711  302  18    3    4 


178  COMPOUND   QUANTITIES. 

Subtract : 

gals.    qts.     pts.  bu.      pks.    qts.  bu.      pks.    qta. 

6.    4     1     0  7.    56     1     0  '  8.    27     1     1 

321  27     3     Q  18     1     3 

9.    Multiply  15  gals.  2  qts.  1  pt.  by  130|. 

10.  Divide  34  bu.  3  pks.  4  qts.  by  9. 

11.  How  many  pint  bottles  will  be  required  to  hold  63 

gals,  of  wine  ? 

12.  How  many  pint  packages  can  a  seedsman  make  from 

3bu.  3  pks.  6  qts.  of  peas? 

13.  In  one  season  a  market-gardener  sold  3758   baskets 

of  strawberries  averaging  1  pt.  each.  How  many 
bushels  did  he  sell  ? 

14.  A  farmer  having  a  flock  of  87  fowls  feeds  them  daily 

2  bu.  1  pk.  1  pt.  of  grain.  What  is  the  average 
amount  for  each  fowl  ? 

15.  A  lady  in  one  month  gave  to  her  4  canaries  f  of  a 

quart  of  seed,  to  her  parrot  2-J-  qts.,  and  to  3  mocking- 
birds If  qts.  How  much  seed  did  she  give  to  all 
the  birds  together  ? 

16.  A  gardener  raised  f  of  a  bushel  of  Lima  beans,  and 

-$•  of  a  bushel  of  Caseknife  beans.  He  sold  3  pks. 
6  qts.  1  pt. ;  how  many  had  he  left  ? 

17.  A  merchant  receives  37  boxes  of  oranges,  amounting 

to  25  bu.  3  pks.  7  qts.  Only  $  of  the  fruit  was  fit 
to  sell ;  how  many  bushels  had  to  be  thrown  away  ? 

18.  A  tank  is  30  ft.  3  in.  long,  16  ft.  4  in.  wide,  and  6  ft. 

4  in.  deep.     Find  how  many  gallons  it  will  hold. 

19.  Find  the  number  of  bushels  in  a  bin  that  is  6  ft.  long, 

5  ft.  wide,  4  ft.  deep.  $ 

20.  How  many  gallons  will  a  cistern  hold  that  is  5  ft. 

square  and  6  ft.  deep? 


COMPOUND   QUANTITIES.  179 

UNITS  OF  WEIGHT. 
Avoirdupois  Weight. 

172.  16  drams  (drs.)        =  1  ounce  (oz.). 

16  ounces  =  1  pound  (lb.). 

100  pounds  =  1  hundred-weight  (cwt.), 

20  hundred-weight  =>  1  ton  (t.). 


112  pounds  =  1  long  hundred- weight. 

2240  pounds  =  1  long  ton. 

NOTE.  Avoirdupois  weight  is  used  for  weighing  all  articles  except 
gold,  silver,  and  jewels. 

In  the  United  States  custom  house  and  in  wholesale  transactions 
in  coal  and  iron  the  long  ton  is  used. 

The  pound  avoirdupois  contains  7000  grains. 

Ex.   114.      (Oral.) 

1.  How  many  ounces  in  2  Ibs.  ?  in  5  Ibs.  ? 

2.  How  many  ounces  in  \  of  a  pound?  in  f  of  a  pound? 

in  -J-  of  a  pound  ?  in  -f-  of  a  pound  ?  in  %  of  a  pound? 

3.  What  part  of  a  pound  are  4  oz.  ?  2  oz.  ?  8  oz.  ?  6  oz.  ? 

12  oz.  ? 

4.  How  many  pounds  in  48  oz.  ?  in  36  oz.  ?  in  24  oz.  ? 

5.  How  many  hundred-weight  in  2  t.  ?  in  3  t.  ? 

6.  How  many  4-oz.  packages  of  nutmegs  can  be  put  up 

from  2-J  Ibs.  of  nutmegs  ? 

7.  If  hay  is  $20  a  ton,  how  many  pounds  can  be  bought 

for  $5?  $7?  $10? 

8.  If  hay  is  $16  a  ton,  what  are  750  Ibs.  worth? 

9.  What  part  of  a  pound  is  $  of  an  ounce  ? 


180  COMPOUND   QUANTITIES. 

10.  If  butter  is  25  cts.  a  pound,  and  hay  is  $16  a  ton,  how 
many  pounds  of  butter  will  it  take  to  pay  for  1500 
Ibs.  of  hay  ? 

Ex.   115. 

1.  Reduce  12,484  oz.  to  higher  denominations. 

2.  Reduce  7  cwt.  64  Ibs.  to  ounces. 

3.  Reduce  95,784  oz.  to  higher  denominations. 

4.  A  bushel  of  wheat  weighs  60  Ibs.     How  many  bushels 

in  lit.? 

5.  A  cubic  foot  of  water  weighs  1000  oz.     In  1800  cu.  ft. 

of  water  how  many  tons  ? 

6.  What  is  the  difference  in  pounds  between  27  long  tons 

of  coal  and  27  short  tons  of  coal  ? 


7.  Find  the  value  of  -^  of  a  ton. 

8.  What  fraction  of  a  pound  is  0.00006  of  a  ton  ? 

9.  Add  |  1.,  £  cwt.,  £  Ib. 

10.  Reduce  8  cwt.  34  Ibs.  to  the  decimal  of  a  ton. 

11.  Find  the  value  of  0.472875  of  a  ton. 

12.  Reduce  12  cwt.  80  Ibs.  6  oz.  to  the  decimal  of  a  ton. 

13.  A   farmer   sold    in    one   week   5.825  t.   of  hay.     On 

Monday  he  sold  1350  Ibs.  ;  on  Tuesday,  •£  t.  ;  on 
Wednesday,  1|-  t.  ;  on  Thursday,  1.415  t.  ;  on  Fri- 
day, If  t.  What  part  of  a  ton  did  he  sell  on 
Saturday  ? 

14.  A  grocer  sold  in  one  day  17  cwt.  83  Ibs.  6  oz.  of  loaf 

sugar,  13  cwt.  95  Ibs.  12  oz.  of  coffee  sugar,  15  cwt. 
78  Ibs.  15  oz.  of  brown  sugar.  Required  the  whole 
amount  sold. 


COMPOUND   QUANTITIES.  181 

15.  A  grocer  has  7  cwt.  57  Ibs.  12  oz.  of  Java  coffee,  5  cwt. 

39  Ibs.  10  oz.  of  Mocha.  After  mixing  the  two 
kinds  of  coffee,  he  sells  from  the  mixture  10  cwt. 
97  Ibs.  9  oz.  How  much  coffee  has  he  left  ? 

16.  A  butcher  receives  from  the  West  every  day,  Sundays 

excepted,  9  cwt.  81  Ibs.  7  oz.  of  beef.  How  much 
does  he  receive  per  week  ? 

17.  A  man  puts  into  his  cellar  17  loads  of  coal,  averaging 

1  t.  387  Ibs.  a  load.     Required  the  whole  amount. 

18.  Divide  19  t.  17  cwt.  58  Ibs.  by  9. 

19.  A  farmer  sells  4  oxen  whose  united   weight  is  2  t. 

7  cwt.  29  Ibs.  13  oz.     What  is  their  average  weight  ? 

20.  Find  f  of  8  t.  16  cwt.  24|  Ibs. 

21.  Divide  15  t.  17  cwt.  29  Ibs.  7  oz.  by  £ 

Troy  Weight. 

173,  24  grains  (grs.)     =  1  pennyweight  (dwt.). 

20  pennyweights  =  1  ounce  (oz.). 
12  ounces  =  1  pound  (lb.). 

NOTE.    Troy  weight  is  used  for  weighing  gold,  silver,  and  jewels. 
The  pound  Troy  contains  5760  grs. 

Ex.   116.     (Oral.) 

1.  How  many  grains  in  2  dwt.?    in  2  dwt.  9  grs.?  in 

3  dwt.  7  grs.  ? 

2.  How  many  pennyweights  in  24  grs.  ? 

3.  How  many  pennyweights  in  1  oz.  ?  in  2  oz.  ?  in  2  oz. 

8  dwt.?  in  5  oz.  17  dwt.? 

4.  How  many  ounces  in  40  dwt.  ?   in  100  dwt.  ?   in  60 

dwt. 


182  COMPOUND   QUANTITIES. 

5.  How  many  ounces  in  1  Ib.  ?   in  5  Ibs.  ?    in  10  Ibs.  ?   in 

3  Ibs.  6  oz.  ?   in  4  Ibs.  9  oz.  ? 

6.  How  many  pounds  in  12  oz.  ?   in  48  oz.  ?   in  72  oz.  ? 

in  80  oz.  ?  in  90  oz.  ? 

7.  If  1  dwt.  of  gold  is  worth  $1.50,  find  the  value  of  1  oz. 

of  gold  ;  1  Ib.  of  gold. 

8.  How  many  spoons  weighing  25  dwt.  each  can  be  made 

from  1  Ib.  3  oz.  of  silver  ? 

9.  If  10  dwt.  of  silver  are  worth  70  cts.,  find  the  value 

of  1  Ib.  of  silver. 

Ex.  117. 

1.  Reduce  3  Ibs.  9  oz.  18  dwt.  17  grs.  to  grains. 

2.  Reduce  25  Ibs.  9  oz.  5  dwt.  to  pennyweights. 

3.  Reduce  3420  dwt.  to  higher  denominations. 

4.  What  is  the  difference  in  weight  between  3  doz.  silver 

tablespoons  weighing  5  Ibs.  9  oz.  8  dwt.  and  3  doz. 
silver  teaspoons  weighing  1  Ib.  9  oz.  16  dwt.  18  grs.  ? 

5.  Required  the  weight  of  8  silver  teapots,  each  weighing 

3  Ibs.  9  oz.  18  dwt.  13  grs. 

6.  When  12  tankards  weigh  36  Ibs.  8  oz.  14  dwt.  16  grs., 

what  is  their  average  weight? 

7.  Find  the  value  of  f  of  a  pound. 

8.  Reduce  -|  of  a  grain  to  the  fraction  of  an  ounce. 

9.  Reduce  7  oz.  10  dwt.  to  the  fraction  of  a  pound. 

10.  Add  0.475  Ibs.,  0.75  dwt.,  0.125  oz,  0.374  Ibs. 

11.  From  0.675  Ibs.  subtract  5.25  oz. 

12.  Reduce  1  oz.  7  dwt.  18  grs.  to  the  decimal  of  a  pound. 


COMPOUND   QUANTITIES.  183 

13.  Reduce  f  dwt.  to  the  fraction  of  a  pound. 

14.  Reduce  4  oz.  4  dwt.  to  the  fraction  of  a  pound. 

15.  What  decimal  of  a  pound  is  -£%  Ib.  —  -|  oz.  ? 

174,    In  preparing  medicines,  apothecaries  use  the  fol- 
lowing : 

Apothecaries1  Weight. 

20  grains  (grs.)  =  1  scruple  (3). 

3  scruples         =  1  dram  ( 3  )• 

8  drams  =  1  ounce  (  §  ). 

12  ounces          =  1  Ib. 

Apothecaries'  Measure. 

60  minims  (fi\,)  =  1  dram  (fl^  lx-)- 

8  drams  =  1  ounce  (fl.  drm.  viij.). 

16  ounces  «=  1  pint  (fl.  oz.  xvj.). 

Ex.  118. 

1.  In  4  Ibs.  854323  how  many  grains? 

2.  In  7864  grs.  how  many  pounds? 

3.  A  patient  is  required  to  take  daily  2  3  2  3  of  bark. 

How  many  weeks  will  7  Ibs.  of  bark  last  him  ? 

4.  Find  the  amount  of  0.4  Ib.  0.25  5    0.375  3  0.648  3 

2.147  grs. 

5.  Subtract  3  5  7  3  12  grs.  from  9  5  6  3  1  3   16  grs., 

and  reduce  the  result  to  the  decimal  of  a  pound. 

6.  How  many  grains  in  1  Ib.  of  apothecaries'  weight? 

7.  What  part  of  a  pound  avoirdupois  is  a  pound  troy  or 

a  pound  apothecaries'  weight  ? 

8.  What  part  of  an  ounce  avoirdupois  is  an  ounce  troy  or 

an  ounce  apothecaries'  weight? 


184  COMPOUND   QUANTITIES. 


UNITS  OF  TIME. 

175,      60  seconds  (sec.)  =  1  minute  (min.). 

60  minutes  =  1  hour  (hr.). 

24  hours  =  1  day  (dy.). 

7  days  -=  1  week  (wk.). 

365  days  (or  52  wks.  1  dy.)  =  1  common  year  (yr.). 

366  days  =  1  leap-year. 
100  years  -=»  1  century. 

The  names  of  the  months  called  calendar  months,  and  the  number 
of  days  in  each  are : 


1.   January  (Jan.)  . 

dy.. 

.     .     .     31 

7.   July    .     .     . 

dy.. 

.     .     31 

2.   February  (Feb.) 
3    March 

.     28  or  29 
31 

8.   August  (Aug.)  . 
9    September  (Sept.)  . 

.     .     31 

30 

4    April 

.     .     .     30 

10    October  (Oct  )  .     . 

.     .     31 

5    May      .... 

.     .     .     31 

11.    November  (Nov.)  . 

.     .     30 

6.    June 

30 

12.   December  (Dec.}    . 

31 

NOTE.  The  number  of  days  in  each  month  may  be  easily  remem- 
bered by  committing  the  following  lines: 

"  Thirty  days  hath  September, 
April,  June,  and  November ; 
All  the  rest  have  thirty-one, 
Except  the  second  month  alone, 
Which  has  but  twenty-eight,  in  fine, 
Till  leap-year  gives  it  twenty-nine." 

A  solar  year  is  365  dys.  5  hrs.  48  min.  50  sec. ;  that  is,  nearly 
365J  days.  As  there  are  365  days  in  a  common  year,  a  common  year 
lacks  nearly  J  of  a  day  of  being  a  solar  year,  and  this  defect  is  made 
up  by  reckoning  for  some  years  (leap-years)  366  days. 

Whenever  the  number  representing  the  year  is  divisible  by  4  and 
not  by  100,  or  is  divisible  by  400,  that  year  is  a  leap-year.  Thus, 
1884,  a  leap-year  ;  1885,  not  a  leap-year;  the  year  1800,  not  a  leap- 
year  ;  the  year  2000,  a  leap-year. 


COMPOUND   QUANTITIES.  185 

Ex.    119.      (Oral) 

1.  How  many  seconds  in  2  min.  ?  in  3  min.  ? 

2.  How  many  minutes  in  2  hrs.  ?  in  3  hrs.  ?  in  60  sec.  ? 

in  120  sec.  ? 

3  How  many  hours  in  2  dys.  ?  in  3  dys.  ?  in  120  min.  ? 
in  360  min.  ? 

4.  How  many  days  from  Jan.  1  to  Feb.  17,  both  days 

inclusive  ? 

5.  How   many   months  from  Aug.   9  to  Nov.   9?  from 

March  5  to  Sept.  5  ?  from  April  4  to  Oct.  4? 

6.  If  a  man  can  do  a  piece  of  work  in  30  min.,  how  many 

hours  will  it  take  him  to  do  four  times  as  much  ? 

7.  If  a  man  can  walk  a  mile  in  15  min.,  how  many  hours 

will  it  take  him  to  walk  24  mi.  ? 

80  At  the  rate  of  3  mi.  an  hour,  how  far  will  a  man  walk 
in  45  min.  ? 

9.  If  a  man  earns  $12  a  week,  and  pays  for  expenses  $12 
per  month  of  4  wks.,  how  much  will  he  save  in 
20  wks.  ? 

10.    If  a  man  walks  -J-  of  a  mile  in  5  min.,  how  many  hours, 
at  that  rate,  will  it  take  him  to  walk  4  mi.  ? 

Ex.  120. 

1.  Reduce  4  yrs.  39  dys.  17  hrs.  22  min.  18  sec.  to  sec- 

onds. 

2.  In  48,967,349  sec.  how  many  years? 

3*  Find  the  exact  length  of  the  lunar  month  which  con- 
tains 2,551,443  sec. 


186  COMPOUND   QUANTITIES. 

4.  How  many  seconds  more  are  there  in  the  3  spring 

months  than  in  the  3  autumn  months  ? 

5.  Reduce  f  of  a  year  to  days. 

6.  Find  the  value  of  0.375  yr.  0.142  dy.  0.27  min. 

7.  What  part  of  a  day  are  12  hrs.  15  min.  25  sec.  ? 

8.  What  part  of  2  dys.  7  hrs.  18  min.  are  1  dy.  3  hrs. 

15  min.  ? 

9.  How  much  greater  is  the  quotient  of  100  yrs.  25  dys. 

12  hrs.  27  min.  28  sec.  divided  by  4  than  the  product 
of  4  yrs.  17  dys.  9  hrs.  12  min.  18  sec.  multiplied 
by  5? 

10.  Find  the  number  of  days,  reckoning  from  noon  of  the 

one  to  noon  of  the  other,  between  Feb.  24  and 
June  23,  1884;  also  between  Dec.  25,  1884,  and 
May  25,  1885. 

11.  How  many  hours  from  noon  of  the  4th  to  midnight  of 

the  7th  of  July,  1885? 

12.  Divide  11  wks.  6  dys.  18  hrs.  by  9. 

13.  Divide  2  yrs.  135  dys.  17  hrs.  by  72. 

14.  From  5  yrs.  17  hrs.  take  2  yrs.  138  dys.  22  hrs. 

15.  Find  the  value  of  3.1725  dys. 

16.  Find  the  value  of  21.325  of  a  year. 

17.  Express  9  dys.  3  hrs.  as  the  decimal  of  a  week. 

18.  Express  13  hrs.  15  min.    17  sec.  as  the  fraction  of 

6  dys.  1  hr.  48  min.  7  sec. 

19.  Express  3  dys.  20  hrs.  35  min.  33  sec.  as  the  decimal 

of  27  dys.  13  hrs.  22  min.  30  sec. 

20.  Find  the  value  of  5.58  yrs. 


COMPOUND   QUANTITIES.  187 


DIFFERENCE  BETWEEN  Two  DATES. 

176,    Find   the   difference   between  April  3,   1885   and 
rs.       mos.  v&.   May  7,  1837. 

K        7  ^p  filing  the  Difference  between  long  dates, 

30  davs  are  considered  a  month.     As  April  is  the 


. 

47  10  2fo  fourth  and  May  the  fifth,  month,  we  write  4  and 
5  instead  of  the  names  of  these  months. 

In  finding  the  difference  between  short  dates,  the  exact  number  of 
days  is  generally  counted. 

Ex.  121. 

1.  On  the  1st  of  January,  1885,  how  much  time  had  passed 

since  the  discovery  of  the  Island  of  San  Salvador, 
Oct.  12,  1492? 

2.  At  the  birth  of  Lafayette,  Sept.  6,   1757,  what  was 

the  age  of  George  Washington,  born  Feb.  22,  1732? 

3.  If  a  note  is  dated  March  5,  1885,  and  has  3  mos.  3  dys. 

to  run,  when  is  the  note  due  ? 

4.  If  a  note  is  discounted  Feb.  1,  and  is  due  April  22,  how 

many  months  and  days  has  it  to  run  ? 
6.    Find  the   exact   number   of  days   from  Sept.  23  to 
Jan.  11. 

NOTE.   In  finding  the  difference  of  these  dates,  the  23d  of 
September  is  not  counted,  but  the  llth  of  January  is. 

6.  Find  the  exact  number  of  days  between  March  5  and 

July  4. 

7.  Find  the  exact  number  of  days  between  June  3  and 

Nov.  1. 

8.  Find  the  exact  number  of  days  between  Feb.  3  and 

June  3,  of  a  common  year. 

9.  Find  the  difference  between  June  7,  1885,  and  July  4, 

1776. 


188  COMPOUND    QUANTITIES. 


ANGULAR  MEASURE. 

177,  A  circle  is  a  plane  figure  bounded  by  a  curved  line 
called  the  circumference,  all  points  of  which  are  equally  dis- 
tant from  a  point  within  called  the  centre.     A  part  of  the 
circumference  is  called  an  arc, 

178,  A  line  drawn  through  the  centre  and  terminated 
by  the  circumference  is  called  a  diameter;   and  half  the 
diameter  is  called  the  radius, 

If  a  straight  line  fixed  at  one  end  is  revolved,  the  other  end  will 
describe  the  circumference  of  a  circle ;  and  the  amount  of  rotation 
of  the  straight  line  from  its  position  at  the  start  to  any  other  given 
position,  is  the  angular  magnitude  described  by  the  moving  straight 
line.  Thus,  if  OA  revolve  about  0  as  a  fixed  point,  the  extremity  A 
will  describe  the  circumference  ABC.  When  OA  has  reached  the 
position  OB,  the  part  of  the  circumference 
between  A  and  B  is  described  by  A,  and 
the  part  of  the  angular  magnitude  about 
the  point  0,  between  OA  and  OB,  is  de- 
scribed by  OA.  The  angle  A  OB  is  such 
a  part  of  the  angular  magnitude  about  0  as 
AB  is  of  the  circumference. 

The  circumference  of  every  circle  is  divided 
into  360  equal  parts,  called  arc-degrees,  and 
corresponding  to  every  one  of  these  equal  parts  is  an  angle  at  the 
centre  of  the  circle.  Hence  the  whole  angular  magnitude  about  any 
point  in  a  plane  is  divided  into  360  equal  parts  called  angle- degrees, 
and  the  number  of  degrees  in  the  angle  formed  by  two  lines  drawn 
from  the  centre  of  a  circle  is  the  same  as  the  number  of  degrees  in 
the  arc  which  is  intercepted  between  these  two  lines. 

179,  An  angle  described  by  a  line  making  one-fourth  of 
a  revolution  contains  90°  and  is  called   a  right  angle,  as 
AOB ;   and  OA  and  OB  are  said  to  be  perpendicular  to 

each  other. 


COMPOUND   QUANTITIES.  189 


UNITS  OP  ANGULAR  MEASURE. 

180,  60  seconds  (")  =  1  minute  ('). 

60  minutes       =  1  degree  (°). 

360  degrees        =  1  revolution. 

NOTE.   A  degree  of  the  circumference  of  the  earth  at  the  equator 
contains  60  geographical  miles,  or  69.16  statute  miles. 


Ex.  122. 

1.  Keduce  49°  37'  29"  to  seconds. 

2.  In  13,978"  how  many  degrees? 

3.  Find  the  value  of  £  of  360°. 

4.  What  part  of  the  whole  angular  magnitude  about  a 

point  is  -|  of  a  second  ? 

5.  Find  the  sum  of  45.425°,  0.115',  0.255". 

6.  Change  0.471  of  a  minute  to  the  decimal  of  a  degree. 

7.  What  part  of  7°  35'  15"  are  3°  20'  45"  ? 

8.  Divide  17°  27'  13"  by  5  ;    multiply  8°  19'  47"  by  8 ; 

and  find  the  difference  between  the  results. 

9.  From  7°  0'  18"  subtract  3°  47'  36". 

10.  The  latitude  of  New  York  is  40°  42'  43"  North ;  the 

latitude  of  Boston  is  42°  21'  30"  North.     Find  their 
difference  in  latitude. 

11.  The  latitude  of  New  Orleans  is  29°  57'  46"  North  ;  the 

latitude  of  Rio  Janeiro  is  22°  56'  South.    Find  their 
difference  in  latitude. 

HINT.    Their  difference  in  latitude  is  found  by  taking  the 
sum  of  their  latitudes. 


190  COMPOUND   QUANTITIES. 


CURRENCY. 

181,  The  coins  of  the  United  States  are :   20-dollar,  10- 
dollar,  5-dollar,  3-dollar,  2^-dollar,  and  1-dollar  gold  coins ; 
1-dollar,  50-cent,  25-cent,  and  10-cent  silver  coins ;  5-cent 
and  3-cent  nickel  coins ;  and  2-cent  and  1-cent  bronze  coins. 

182,  As  any  sum  of  money  can  be  expressed  in  United 
States  currency  as  dollars  and  decimal  fractions  of  a  dollar, 
it  is  always  best  to  treat  United  States  money  as  a  simple 
quantity. 

183,  The  same  is  true  of  French,  Italian,  German,  Rus- 
sian, and  Austrian  currency. 

184. 

French  Currency :  100  centimes  -  1  franc  (fr.)  =  $0.193 
Italian  Currency  .  100  centissimi  =  1  lira  =  $0.193. 

German  Currency :  100  pfennigs  =  1  mark  =  $0.238. 
Russian  Currency  .  100  kopecks  «-  1  rouble  =  $0.734. 
Austrian  Currency  :  100  kreutz ere  «=  1  florin  (fl.)  -  $0.453. 

English  Currency. 

185,  4  farthings  =  1  penny  (d.). 

12  pence       =  1  shilling  (s.). 
20  shillings  =  1  pound  (£). 

A  guinea       =  21 «.  A  crown  =  5  s. 

A  sovereign  =  20  s.  A  florin   =  2  s. 

A  sovereign  =  $4.866J. 

Ex.  123.     (Oral.) 

1.  How  many  shillings  in  48  d.  ?  in  60  d.  ? 

2.  How  many  pence  in  5s.  ?  in  10s.?  in  a  sovereign? 

3.  How  many  shillings  in  £3£?  in  £2f  ? 


COMPOUND   QUANTITIES.  191 

4.  How  many  pounds  in  95  s.  ?  in  100  5.  ? 

5.  How  many  pence  in  a  crown  ?  in  a  florin? 

6.  How  many  shillings  in  a  guinea  ?   in  a  half-sovereign  ? 

7.  What  part  of  a  pound  are  4  s.  ?  5s.?  8s.? 

8.  What  part  of  a  shilling  are  6  d.  ?  4  d.  ?  3  rf.  ? 

9.  How  many  pence  in  1  s.  3  d.  ?  in  2  s.  6  d.?  in  3  s.  2  c?.V 

Ex.  124. 

1.  Reduce  £432  15s.  10  d.  to  pence. 

2.  Change  4238  farthings  to  higher  denominations. 

3.  Express  in  dollars  the  value  of  £18. 

4.  Express  in  English  money  $60.83. 

5.  Express  in  United  States  money  £3  16  s. 

6.  How  many  sovereigns  are  equal  in  value  to  $389.32  ? 

7.  Reduce  £  s.  to  the  fraction  of  a  guinea. 

8.  What  part  of  13s.  2 d.  1  farthing  are  9s.  10 d.  2  farthings? 

9.  Find  the  value  of  £5.375. 

10.  Reduce  6s.  5  d.  3.04  farthings  to  the  decimal  of  a  pound. 

11.  Express  in  pounds  £5  9  s.  3  d. 

12.  Add  £0.75,  0.125  guineas,  0.54s.,  0.55  d. 


MISCELLANEOUS. 
186, 

Numbers. 

12  units  =  1  dozen. 
12  dozen  =  1  gross. 
12  gross   =  1  great  gross. 
20  units    =  1  score. 


Paper. 

24  sheets     =  1  quire. 
20  quires     =  1  ream. 

2  reams     =  1  bundle. 

5  bundles  =  1  bale. 


192  COMPOUND  QUANTITIES. 


•tnn  Weights. 


A  bushel  of  corn  or  rye  =  56  Ibs. 
A  bushel  of  corn  meal,  -\ 

rye  meal,  or  cracked  >  =  50  Ibs. 


corn, 


) 


A  bushel  of  wheat  =  60  Ibs. 

A  bushel  of  potatoes  =  60  Ibs. 

A  bushel  of  beans  =  60  Ibs. 

A  bushel  of  oats  =  32  Ibs. 


A  bushel  of  barley     =  48  Ibs. 


i 


.  45  Ibs. 


-  14  Ibs. 


A  bushel  of  timo- 
thy-seed 

A  stone  of  iron  or 
lead 

A  pig  of  iron  or  lead  =  21J  stone. 

A  fother  of  iron  or  )   =  g    • 
lead  J 


The  weight  of  a  bushel  of  potatoes,  corn,  etc.,  varies  slightly  in 
different  States,  but  the  weights  here  given  are  those  generally 
adopted  in  business  transactions. 

A  barrel  of  flour  =  196  Ibs. 

A  barrel  of  pork  or  beef  =  200  Ibs. 
A  cask  of  lime  =  240  Ibs. 

A  cental  of  grain  =  100  Ibs. 

A  quintal  of  fish  =  100  Ibs. 

Books. 
188,    A  book  formed  of  sheets  folded  in 

2  leaves  is  a  folio  ; 

4  leaves  is  a  quarto  ; 

8  leaves  is  an  octavo  ; 
12  leaves  is  a  duodecimo  ; 
16  leaves  is  a  16mo. 

Ex.  125. 

1.  How  many  barrels  in  75  t.  of  beef? 

2.  In   a   car-load   of  36,000   Ibs.  of  wheat,  how   many 

bushels  ? 

3.  Find  the  weight  of  27  bu.  of  potatoes. 

4.  How  much  paper  will  be  used  by  an  author  who  sends 

to  a  semi-weekly  paper  6  sheets  of  manuscript  twice 
a  week  for  a  year  ? 


COMPOUND    QUANTITIES. 


193 


5.  Reduce  -J  of  a  quire  to  the  fraction  of  a  bundle. 

6.  Reduce  2  bundles  6  quires  6  sheets  to  the  fraction  of 

2  bales  1  bundle. 

7.  Reduce  3  bundles  7  quires  18  sheets  to  the  decimal 

fraction  of  a  bale. 

8.  A  button  manufactory  makes  96  dozen  buttons  a  day. 

How  many  great  gross  will  it  make  in  24  wks.  ? 

9.  Find  the  weight  of  103  bu.  3  pks.  4  qts.  of  barley. 
10.    In  5  t.  624  Ibs.  of  oats,  how  many  bushels? 


LONGITUDE  AND  TIME. 

189,  A  meridian  is  any  line  drawn  straight  around  the 
earth,  and  passing  through  both  poles. 

190,  The  longitude  of  a  place  is  the 
angle  of  inclination  of  the  two  planes 
which  are  supposed  to  pass  through 
the  centre  of  the  earth,  and  contain, 
the  one  the  meridian  of  that  place, 
and  the  other  the  standard  meridian. 
Thus,  the  longitude  of  (7,  reckoned 
from  meridian   ABE,   is  the   angle 

BOO,  OB  and  GO  being  both  perpendicular  to  the  diam- 
eter of  the  earth  AE  at  the  point  0.  Places  on  the  Eastern 
Hemisphere  are  in  East  Longitude ;  on  the  "Western  Hem- 
isphere, in  West  Longitude. 

191,  As  the  earth  turns  upon  its  axis  once  in  twenty- 
four  hours,  a  point  on  the   earth's  surface  will   describe 
a  circumference  (360°)  in  twenty-four  hours.       Therefore 
longitude  may  be  reckoned  in  time  as  well  as  in  degrees. 

In  one  hour  a  point  on  the  earth's  surface  describes  -fa  of 


194  COMPOUND   QUANTITIES. 

360°  =  15°  ;    in  one  minute  -fa  of  15°  =  15' ;    and  in  one 
second  ^  of  15' =15". 

Again,  since  it  requires  one  hour  (60  min.)  for  a  point  to 
pass  over  15°,  to  pass  over  1°  it  requires  -fa  of  60  min.  =  4 
min. ;  and  to  pass  over  1'  it  requires  -fa  of  4  min.  =  4  sec. 

192,  Express  20°  36'  15"  of  longitude  in  time. 

15)  20°  36'  15" Since  15°  longitude   give  1   hr.   in 

Ihr.  22  min.  25  sec.      time>  I5f  longitude   1   min.,   and   15" 
longitude  1  sec.,  divide  20°  36'  15"  by 

15,  as   in  compound  division,  and  the  quotient  will  be  the  time 
required. 

193,  Express  1  hr.  4  min,  4  sec.  in  degrees. 

1  hr.  4  min.  4  sec.        Since  1  hr.  of  time  equals  15°  of  longitude, 
15  1  min.  of  time  15f,  and  1  sec.  of  time  15",  mul- 

16°  1'  0"  ^Pty  1  kr.  ^  m^n-  4  8ec-  hy  15,  as  in  compound 

multiplication,  and  the  product  will  be  the  lon- 
gitude required. 

194,  Hence,  if  longitude  is  expressed  in  degrees,  divide  by 
15  ;  the  quotient  gives  the  longitude  in  hours,  minutes,  and 
seconds. 

195,  If  longitude  is  expressed  in  time,  multiply  by  15 ; 
the  product  gives  the  longitude  in  degrees,  minutes,  and 
seconds. 

Ex.  126. 

1.  The  difference  in  time  between  New  York  and  Paris 

is  5  hrs.  5  inin.  20  sec.     What  is  the   difference  in 
longitude  ? 

2.  Boston  is  71°  3'  and  San  Francisco  122°  26'  west  of 

Greenwich.     What  is  the   difference  in  clock-time 
between  the  two  cities  ? 


COMPOUND   QUANTITIES.  195 

3.  The  difference  in  clock-time  between  New  York  arid 

Canton  is  12  hrs.  28  min.  12  sec.     Find  the  differ- 
ence in  longitude. 

4.  The  difference  in  longitude  between  Cincinnati  and 

Boston  is  13°  26'.     Find  the  difference  in  time. 

5.  New  York  is  74°  and  Cincinnati  is  84°  30'  west  longi- 

tude.    Find  the  difference  in  time. 

6.  The  difference  in  time  between  Canton  and  Cincinnati 

is   13  hrs.  10  min.  8  sec.     Find  the  difference  in 
longitude. 

7.  The  difference  in  longitude  between  New  York  and 

Canton  is  187°  3'.     What  is  the  difference  in  time  ? 

8.  Find  the  difference  in  time  between  Philadelphia,  lon- 

gitude 75°  10f  West,  and  Buenos  Ayres,  longitude 
58°  22f  West. 

9.  The  difference  of  time  between  St.  Petersburgh  and 

New  Orleans  is  8  hrs.  1  min.  16  sec.     What  is  the 
difference  in  longitude  ? 

10.  Find  the  difference  in  time  between  the  Cape  of  Good 
Hope,  longitude  18°  28'  East,  and  Halifax,  longitude 
63°  36' West? 

196,  Since  the  sun  appears  to  move  from  east  to  west, 
sunrise  will  occur  earlier  at  all  points  east,  and  later  at  all 
points  west,  of  a  given  place.  Hence,  clock-time  will  be 
later  in  all  places  east,  and  earlier  in  all  places  west,  of  a 
given  meridian. 

Therefore,  if  the  time  of  a  place  be  given, 

To  find  the  time  of  a  place  east,  add  to  the  given  time 
the  difference  of  time  between  the  two  places. 

To  find  the  time  of  a  place  west,  subtract  from  the  given 
time  the  difference  of  time  between  the  two  places. 


196  COMPOUND   QUANTITIES. 


To  FIND  THE  DIFFERENCE  IN  CLOCK-TIME  WHEN  THE  DIFFERENCE 
IN  LONGITUDE  is  KNOWN. 

When  it  is  noon  at  Boston  (long.  71°  3f  30"  West),  what 
is  the  time  at  Paris  (long.  2°  20'  22"  East)  ? 
71°    3f30"W. 
2°  20'  22"  E. 
73°  23'  52"  .  .  .  difference  in  longitude. 

15)  73°  23'  52" 

4  hrs.  53  min.  35.47  sec. 

G  min.  24}  sec.  before  5  P.M.  Ans. 

Since  Boston  is  west  and  Paris  is  east  of  the  meridian  of  Green- 
wich, the  difference  between  their  longitudes  is  found  by  taking  the 
sum  of  their  longitudes. 

Their  difference  in  longitude,  73°  23'  52",  is  equivalent  to  4  hrs. 
53  min.  35.47  sec.,  and  as  Paris  is  east  of  Boston,  the  time  at  Paris  is 
found  by  adding  the  4  hrs.  53  min.  35.47  sec.  to  the  time  at  Boston. 

Ex.  127. 

1.  When    it  is    noon   at  Chicago,  what  is  the  hour  at 

New  York,  the  difference  in  longitude  being  13°  37'? 

2.  What  is  the  time  in  London  when  it  is  half-past  3  in 

the  afternoon  at  Constantinople,  Constantinople  be- 
ing 29°  east  of  London  ? 

3.  The  longitude  of  New  York  is  74°  West,  that  of  Paris 

is  2°  20'  East,     When  it  is  15  minutes  past  10  A.M. 
in  New  York,  what  is  the  time  in  Paris? 

4.  The  longitude  of  Boston  is  71°  3'  and  that  of  New 

York  74°  West.     What  is  the  time  in  Boston  when 
it  is  midnight  in  New  York  ? 

5.  The  difference  in  longitude  between  San  Francisco  and 

Chicago  is  34°  49f.    What  time  is  it  at  San  Francisco 
when  it  is  9  o'clock  P.M.  at  Chicago? 


COMPOUND   QUANTITIES  197 


6.  Paris  is  45°  10'  east  of  Rio  Janeiro.     What  time  is  it 

at  Rio  Janeiro  when  it  is  7  o'clock  P.M.  at  Paris? 

7.  If  the  sun  rises  at  half-past  4,  when  it  is  sunrise  at 

Richmond,  Va.,  what  is  the  time  at  Rouen,  France, 
the  difference  of  longitude  being  78°  46'  ? 

8.  The  French  residents  in  Calcutta  wish  to  unite  with 

the  people  of  Paris  in  a  celebration  to  occur  at 
3  o'clock  P.M.  Paris  is  2°  20'  East,  Calcutta, 
88°  27 '  East.  At  what  hour  must  the  festivities 
begin  in  Calcutta  ? 

NOTE.  Standard  Time  is  the  clock-time  of  some  selected  meridian. 
Eastern  standard  time  is  the  clock- time  of  the  meridian  75°  west  of 
Greenwich,  and  is  five  hours  slow  of  Greenwich  time.  Central 
standard  time  is  the  clock-time  of  90°  west  of  Greenwich,  and  is 
just  one  hour  slow  of  Eastern  standard  time.  Mountain  standard 
time  is  the  clock-time  of  the  meridian  of  105°,  and  is  one  hour  slower 
than  that  of  90°.  Western  standard  time  is  the  clock-time  of  the 
meridian  of  120°,  and  is  one  hour  slower  than  that  of  105°.  The 
railroads  and  many  cities  and  towns  of  the  United  States  have 
adopted  standard  time. 

MISCELLANEOUS  EXAMPLES. 

Ex.  128. 
1.    Find  the  amount  of  the  following  bill : 


/f       >3    -M-ftd.     dfls£tsM-         (cb. 
7 

@ 
® 


How   much    change   out   of  a   20-dollar  bill  should  the 
purchaser  receive? 


198  COMPOUND  QUANTITIES. 

Make  out  the  bills  for  : 

2.  27    yds.  of  flannel       at    80  cts.  a  yard  ; 
32    yds.  of  calico         at    11  cts.  a  yard  ; 

3£  doz.  of  stockings  at    $2  per  dozen  ; 
6    pairs  of  gloves      at    84  cts.  a  pair  ; 

4  collars  at    35  cts.  each. 

3.  10    Ibs.  of  sugar  @    10    cts.  a  pound  ; 

6    Ibs.  of  tea  @    88    cts.  a  pound  ; 

8  Ibs.  of  coffee  @  32  cts.  a  pound  ; 
12  Ibs.  of  currants  @  11-J-  cts.  a  pound  ; 

10  Ibs.  of  rice  @  9  cts.  a  pound. 

4.  18|  Ibs.  of  beef  @    22    cts.  a  pound  ; 
10^  Ibs.  of  mutton  @    21    cts.  a  pound  ; 

7-J-  Ibs.  of  pork  @    17    cts.  a  pound  ; 

16    Ibs.  of  veal  @    16    cts.  a  pound  ; 

14f  Ibs.  of  ham  @    20    cts.  a  pound. 

5.  5£  Ibs.  of  soap  @      9^  cts.  a  pound  ; 
3^-  Ibs.  of  candles  @    13    cts.  a  pound  ; 
2    Ibs.  of  butter  @    35    cts.  a  pound  ; 

56    Ibs.  of  rice  @      4^-  cts.  a  pound. 

6.  7    doz.  and  4  eggs  @     18    cts.  per  dozen  ; 
19    Ibs.  of  soap  @    11    cts.  per  pound  ; 
18    Ibs.  of  butter  @    28    cts.  per  pound ; 
13^-  Ibs.  of  cheese  @    15    cts.  per  pound  ; 

^  Ib.  pepper  @      2^-  cts.  per  ounce. 

7.  12|  t.  of  hay  @    $18      a  ton  ; 
66    bu.  of  rye  @    $1.26    a  bushel ; 

102    bu.  of  barley  @    78  cts.  a  bushel  ; 

5  bbls.  of  flour  @    $6.60   a  barrel. 


COMPOUND   QUANTITIES.  199 

8.  A  man  walks  1  mi.  47  rds.  in  20  min.     How  many 

hours  will  it  take  him  to  walk  41  mi.  92  rds.  ? 

9.  Kequired  the  cubic  feet  of  a  box  6  ft.  6  in.  long,  4  ft. 

9  in.  wide,  and  3  ft.  3  in.  deep. 

10.  What  will  be  the  weight  of  a  wall  of  brick- work  10  ft. 

long,  1-J-  ft.  thick,  4£  ft.  high,  if  each  cubic  foot 
weighs  120  Ibs.  ? 

11.  How  many  cubic  yards  of  earth  will  be  cut  out  of  a 

drain  420  ft.  long,  2  ft.  wide,  and  4  ft.  deep  ? 

12.  What  will  be  the  expense  of  glazing  a  window  of  16 

squares,  each  LJ-  ft.  long  and  -f  ft.  wide,  at  $1.08 
per  square  foot  ? 

13.  What  length  must  be  cut  off  an  inch-board  9  in.  wide 

to  obtain  4  ft.  board  measure  ? 

14.  How  many  boards  each  11^-  ft.  long,  and  10  in.  wide 

will  be  required  for  the  flooring  of  a  room  23  ft.  long 
and  17  ft.  Gin.  wide? 

1.5.  A  farm  of  22^-  acres  is  divided  into  house-lots  measur- 
ing 75  yds.  in  length  by  33  yds.  in  breadth.  How 
many  lots  are  there  ? 

16.  At  9  cts.  per  cubic  foot,  what  will  be  the  cost  of  a 

block  of  stone  9  ft.  long,  5^-  ft.  wide,  and  4  ft.  thick? 

17.  At  50  cts.  an  ounce,  what  is  the  value  of  a  silver  cup 

weighing  15  oz.  12  dwt.  12  grs.  ? 

18.  If  the  cost  of  making  a  barrel  of  flour  into  bread  is 

$2.20,  and  flour  is  worth  $9  a  barrel,  what  should  a 
baker  receive  for  a  loaf  containing  If  Ibs.  of  flour  ? 

19.  At  $30  per  M.,  what  is  the  value  of  a  stick  of  timber 

24  ft.  long,  and  2  ft.  square  at  the  end  ? 


200  COMPOUND   QUANTITIES. 

20.  A  schoolroom  is  44  ft.  long,  28-^  ft.  wide,  and  13  ft. 

high.  What  will  be  the  cost  of  painting  the  four 
walls  and  the  ceiling,  at  the  rate  of  18  cents  a  square 
yard,  making  no  allowance  for  doors  and  windows  ? 

21.  A    druggist   pays   50   cts.    a   pound   avoirdupois  for 

chloride  of  potash,  and  retails  it  in  powders  contain- 
ing 135  grs.,  at  5  cts.  each.  How  much  will  he 
gain  on  5£  Ibs.  ? 

22.  Find  the  entire  surface  of  a  block  of  marble  8  ft.  long, 

2  ft.  wide,  1£  ft.  thick. 

23.  How  many  revolutions  will  be  made  by  a  wheel  3| 

yds.  in  circumference  in  passing  over  198  mi.  ? 

24.  When  an  ounce  of  gold  is  worth  $16.25,  what  must  be 

paid  for  -^  of  a  pound  ? 

25.  If  candles  8£  in.  long  are  worth  9  cts.  a  half-dozen, 

and  candles  10J  in.  long  are  worth  11  cts.  a  half- 
dozen,  which  is  the  better  kind  to  buy  ? 

26.  How  many  silver  spoons  weighing  1  oz.  18  dwt.  12 

grs.  each  can  be  made  from  23  oz.  2  dwt.  of  silver  ? 

27.  An  apprentice  14  yrs.  11  mos.  14  dys.  old  is  to  serve 

his  employer  until  he  is  21  yrs.  of  age.  How  long  is 
he  to  stay  with  his  employer  ? 

28.  What  is  the  rate  per  hour  of  a  horse  that  travels  18  mi. 

1620  yds.  in  3  hrs.  45  min.  ? 

29.  «At  15  cts.  a  yard,  what  will  be  the  cost  of  fencing  a 

rectangular  field  325  yds.  long  and  215  yds.  wide  ? 

30.  What  will  be  the  width  of  carpeting,  if  120  yds.  are 

necessary  to  cover  a  floor  30  ft.  long  and  22-J-  ft. 
wide? 


COMPOUND    QUANTITIES.  201 

31.  When  the  mercury  in  the  tube  of  a  barometer  is  30  in. 
high,  the  pressure  of  the  atmosphere  is  about  15  Ibs. 
for  every  square  inch.  What  will  be  its  pressure 
when  the  mercury  stands  25  in.  high? 

32-  A  cistern  containing  60  hhds.  of  water  has  two  pipes 
open,  by  one  of  which  3  gals,  of  water  per  minute 
run  in,  and  by  the  other  9  gals,  run  out.  In  how 
many  hours  will  the  cistern  be  emptied  ? 

33.  How  many  pounds  of  cement  will  be  required  to  plas- 

ter an  open  cistern  whose  dimensions  are  4^-  it. 
long,  3^-  ft.  wide,  and  2|  ft.  deep,  if  the  cement  on 
a  square  foot  weighs  6 J  Ibs.  ? 

34.  How  many  tons  of  water  will  the  cistern  in  example 

33  hold,  if  a  cubic  foot  of  water  weighs  1000  oz.  ? 

35.  What  will  be  the  cost  of  covering  with  paper  %  yd. 

wide  the  four  walls  of  a  room  21  ft.  long,  16  ft.  wide, 
and  10  ft.  high,  if  the  cost  of  the  paper  is  12£  cts. 
per  yard,  and  no  allowance  is  made  for  doors  and 
windows  ? 

36.  What  is  the  breadth  of  a  rectangular  field  containing 

7£  A.,  if  the  length  is  242  yds.  ? 

37.  A  certain  watch  gains  3|-  sec.  in  24  hrs.,  and  another 

loses  2-z1-  sec.  in  the  same  time.  If  both  be  set  right 
on  Monday  at  noon,  what  will  be  the  difference 
between  them  at  6  o'clock  (true  time)  the  next 
Saturday  evening  ? 

38.  A  milkman  paid  a  farmer  $3.20  for  ten  2-gal.  cans  of 

milk.  He  lost  20  qts.  At  what  price  per  quart 
must  he  retail  the  remainder  to  gain  8  cts.  a  gallon? 

39.  A  man  who  had  f  of  a  square  mile  of  woodland  sold 

12^-  sq.  rds.     How  much  had  he  left  ? 


202  COMPOUND  QUANTITIES. 

40.  How  many  yards  of  Florence  silk  -|  yd.  wide  will  be 

required  to  line  19  yds.  of  camel's  hair  cloth  1-j-  yds. 
wide  ? 

41.  How  many  days  from  Sept.  16,  1882,  to  Feb.  12,  1884? 

42.  A  miller  makes  154  bbls.  of  flour  from  885£  bu.  of 

wheat.  How  many  bushels  on  the  average  are 
required  for  a  barrel  of  flour  ? 

43.  What  must  be  the  length  of  a  walk  2-J-  ft.  wide  to  con- 

tain 38  sq.  ft.  ? 

44.  A  cistern  7  ft.  long  and  5  ft.  wide  contains  105  cu.  ft. 

What  is  its  depth  ?  how  many  gallons  of  water  will 
it  hold  ? 

45.  What  must  be  paid  for  a  pile  of  wood  25  ft.  long,  3  ft. 

high,  and  4  ft.  wide  at  $5.50  per  cord? 

46.  Each  person  on  the  average  breathes  28  cu.  ft.  of  air 

in  an  hour.  How  many  hours  will  the  air  in  a  room 
14  ft.  long,  12  ft.  wide,  and  6  ft.  high  last  12  men  ? 

47.  Sound  travels  at  the  rate  of  1130  feet  a  second.     How 

long  after  the  flash  will  the  clap  of  thunder  come 
when  the  cloud  is  2  mi.  1000  yds.  distant  ? 

48.  There  are  9  oz.  of  iron  in  the  blood  of  1  man.     How 

many  men  would  furnish  iron  enough  in  their  veins 
to  make  a  ploughshare  weighing  22|  Ibs.  ? 

49.  The  fore  wheel  of  a  carriage  is  4  ft.  7  in.  in  circum- 

ference, and  the  hind  wheel  5  ft.  6  in.  How  many 
more  times  will  the  fore  wheel  turn  than  the  hind 
wheel  in  going  a  distance  of  1  mi.  ? 

50.  A  boarding-house  uses  3  pks.  of  potatoes  daily.     At 

87-J-  cts.  per  bushel,  what  will  be  the  expense  for  pota- 
toes during  October,  November,  and  December  ? 


CHAPTER  X. 

PERCENTAGE. 

197,  1.  A  boy  gave  away  6  marbles  out  of  every  hundred 
he  had.  How  many  did  he  give  away  out  of  400? 
of  600?  of  1000? 

2,  A  man  had  300  sheep,  and  sold  10  out  of  every  hun- 

dred.    How  many  did  he  sell  ? 

3,  A  man  sold  7  tons  of  coal  out  of  every  hundred  he  had. 

How  many  did  he  sell  out  of  900  tons  ? 

4,  A  man  sells  11  Ibs.  of  sugar  out  of  every  hundred  he 

has.     How  many  pounds  will  he  sell  out  of  500? 
How  many  pounds  will  he  have  left? 

198,  In  considering  the  increase  or  decrease  of  quantities, 
we  usually  employ  the  number  100  as  the  representative  of 
the  quantity  considered, 

199,  Instead  of  using  the  phrases  6  in  every  hundred,  10 
in  every  hundred,  7  in  every  hundred,  11  in  every  hundred, 
we  say  6  per  cent,  10  per  cent,  7  per  cent,  11  per  cent.     The 
words  per  cent  therefore  mean  hundredths. 

200,  The  symbol  %  is  used  for  the  words  per  cent. 

How  many  hundredths  of  a  number  are  : 

20%? 
25%? 


204  PERCENTAGE. 


How  many  per  cent  of  a  number  is  : 

0.20?  0.75?  0.12J?  1.40? 

0.15?  0.06^?  0.50?  2.25? 

What  common  fraction  of  a  number  (in  its  lowest  terms)  is  • 

10%?          25%?  8|%?         12|%?          125%? 

20%?          50%?  6J%?         66|%?          160%? 

16|%?          75%?         334%?         100%?          175%? 

Express  as  hundredths  and  as  per  cent  : 

i;      I'      *;       f:      A;      A;       A; 

8  >  S  »  «  >  TS"  »  TO"  '»  SO  i 

4  »  5>  T  i  Toi  Ai  ^5' 


201.    Express  \  %  as  hundredths  and  as  a  common  frac- 


tion. 


In  like  mariner  express  as  hundredths  and  also  as  com- 
mon fractions  - 


tyo  :         I7o  ;         f*  ;         f£>  ; 

3  0£   •  _1_  fij.  .  5  C/n  •  JL  0£)  • 

£%;       1%;       1%;       1%;       •&%• 

202,   Find  8%  of  250  bu.  of  corn. 

8%  of  a  number  is  yj^  of  the  number  ;  and  T|^  of  250  bu.  =•  20  bu. 

20  bu.  Ans. 
Find  20%  of  80  yds.  of  cloth. 

20%  =  iSfr  =  i  J  and  J  of  80  yds.  is  16  yds. 

16  yds.  Ans. 


PERCENTAGE. 


205 


Ex.  129.     (Oral) 


8.  80%  of  400  A. 

9.  6J%  of  320  rds. 

10.  121%  Of  400  melons. 

11.  66f%  of  300  oranges. 

12.  8-^%  of  1  doz.  eggs. 


Find: 

1.  4%  of  400  sheep. 

2.  5%  of  1000  bricks. 

3.  8%  of  200  ft.  of  board. 

4.  6%  of  90  dys. 

5.  10%  of  150  cds.  of  wood. 

6.  20%of250prs.ofgloves.      13.    75%  of  40  hens. 

7.  25%  of  120  horses.  14.    60%  of  20  girls. 

15.  16|%  of  60  Ibs.  of  butter. 

16.  37|%  of  120  gals,  of  syrup. 

17.  62|%  of  800  soldiers. 

18.  |%  of  500  bu.  of  wheat. 

19.  \ r%  of  4000  yds.  of  cloth. 

20.  |%  of  100  dollars. 


Ex.  130. 


Find  : 

1.  9%  of  1297. 

2.  2|%  of  4300. 

3.  |  of  1%  of  1346. 

4.  12%  of  6072. 


5.  1%  of  150,975. 

6.  1TV%  of  1984. 

7.  150%  of  1050. 

8.  100%  of  7968. 


9.  A  farmer  having  a  flock  of  1200  sheep  lost  37%  of 
them.  What  per  cent  of  them,  and  how  many  sheep, 
had  he  left  ? 

10,    If  copper  ore  yields  6%  of  pure  metal,  how  many 
pounds  of  copper  will  be  obtained  from  1  t.  of  ore  ? 


206  PERCENTAGE. 


11.  If  a  man  buys  24  A.  of  land  at  $84  an  acre,  what 

must  be  the  annual  income  that  the  investment  may 
yield  10%  ? 

12.  A  grocer  bought  40  cwt.  of  sugar  for  $240.     4%  of 

it  is  wasted,  and  the  remainder  is  retailed  so  that 
there  is  neither  loss  nor  gain.  What  is  the  retail 
price  per  pound  ? 

13.  A  stone-mason  contracted  to  dig  a  cellar  45  ft.  long, 

36  ft.  wide,  and  6  ft.  deep  at  25  cts.  a  cubic  yard, 
He  lost  5%  of  his  contract  price.  What  was  his 
loss? 

14.  A  coal-dealer  bought  25,784  t.  of  coal  at  $5  a  ton. 

He  sold  40%  of  it  at  $7,  20%  at  $8.50,  and  the 
remainder  at  $4.50.  How  much  did  he  gain? 

15.  A  gentleman  owns  2  farms.    The  first  contains  360  A., 

and  the  number  of  acres  in  the  second  is  150%  of 
the  number  of  acres  in  the  first.  Find  the  number 
in  the  second  farm. 

203,  What  per  cent  of  20  is  5  ? 
5  =  &or  Jof20;  and  J  - -flfr  =  25%. 
Therefore  5  =  25%  of  20.  25  % .     Ans. 

Ex.  131.     (Oral.) 
What  per  cent  of: 

1.  16  is  8?  9.  $72  are  $18? 

2.  20  is  5?  10.  $52  are  $39? 

3.  25  is  15  ?  11.  50  qts.  are  5  qts.  ? 

4.  48  is  8?  12.  66  gals,  are  6  gals.  ? 

5.  100  is  12£?  13.  480  dys.  are  24  dys.  ? 

6.  2is£?  14.  90  cds.  are  9  cds.  ? 

7.  3  is  |?          8.   fisf?  15.  80  men  are  50  men  ? 


PERCENTAGE.  207 


Ex.  132. 

1.  From  a  school  of  150  scholars,  50  are  absent.     What 

per  cent  of  the  whole  is  the  number  present  ? 

2.  In  a  school  numbering  200  the  daily  average  atten- 

dance is  160.  What  is  the  per  cent  of  attendance  ? 
The  number  absent  on  the  average  is  what  per  cent 
of  the  number  present  ? 

3.  A  person  bought  a  house  and  lot  for  $6000,  paying 

$5000  for  the  house.  The  value  of  the  lot  is  what 
per  cent  of  the  value  of  the  house  ? 

4.  From  a  peck  of  corn  a  crop  of  48J  bu.  was  raised. 

What  per  cent  was  the  increase  ? 

5.  From  67-^-  bu.  of  corn,  6  bu.  3  pks.  are  sold.     What 

per  cent  of  the  whole  is  sold  ? 

6.  A  house  worth  $8000  rents  for  $720  a  year.     What 

per  cent  of  its  value  does  it  rent  for  ? 

7.  From  Delhi  to  Bombay  the  distance  is  720  miles,  and 

from  Delhi  to  Madras  1080  miles.  What  per  cent 
of  the  distance  to  Madras  is  the  distance  to  Bombay  ? 

8.  Westminster  Hall  is  270  ft.  long  and  75  ft.  broad. 

What  per  cent  of  the  length  is  the  breadth? 

9.  The  Peak  of  Teneriffe  is  12,232  ft.  high.     What  per 

cent  of  a  mile  is  its  height  ? 

10  The  Danube  is  1630  miles  long,  and  the  Missouri  from 
its  source  to  the  Gulf  of  Mexico  is  4000  miles  long 
What  per  cent  of  the  length  of  the  Missouri  is  the 
length  of  the  Danube  ? 

11.  What  per  cent  of  7  hrs.  and  30  min.  are  6f  min.  ? 

12.  What  per  cent  of  3  wks.  and  4  dys.  are  3  dys.  and 

10    hrs.  ? 


208  PERCENTAGE. 


204,   What  is  the  number  of  which  15  is  5%  ? 
If  15  is  5%,  then  15  is  y^y  or  ^V  of  the  number,  and,  if  15  is  -fa  of 
the  number,  the  number  itself  will  be  20  x  15  =  300. 

300.  Ans. 
Ex.  133.     (Oral.) 
What  is  the  number  of  which  : 

1.  10  is  20%?          6.  |is!6£%?  11.  J  is  175%? 

2.  3  is  10%?  7.  f  is  50%?  12.  17  is  34%  ? 

3.  8  is  25%  ?  8.  f  is  75%  ?  13.  50  is  62£%  ? 

4.  4  is  6£%?  9.  60  is  60%?  14.  300  is  0.3%? 

5.  5  is  8£%  ?          10.  50  is  40%  ?  15.  20  is 


Ex.  134. 

1.  10.08  is  16%  of  what  number? 

"ps   2.  24  is  7|%  of  what  number? 

3.  10.94  is  i%  of  what  number  ? 

4.  2500  is  12|%  of  what  number? 

5.  960  is  33£%  of  what  number  ? 

6.  6000  is  20%  of  what  number? 
~f   7.  990  is  110%  of  what  number? 

8.  810  is  90%  of  what  number? 

9.  980  is  175%  of  what  number? 

10.  A  city  in  5  yrs.  increased  12,000  in  population,  a  gain 

of  25%.    What  was  the  population  at  the  beginning 
and  end  of  the  5  yrs.  ? 

11.  A  schoolboy  in  one  week  read  450  lines   of  Latin, 

which  was  75%  of  the  number  in  the  book.     How 
many  lines  had  he  still  to  read  ? 


PERCENTAGE.  209 


12.  A  boy  sold  chestnuts  at  12-J-  cts.  a  quart,  which  was 

200%  of  their  cost.    What  did  they  cost  a  bushel? 

13.  A  clerk  spent  60%  of  his  salary  for  board,  20%  of  it 

for  clothes,  11%  for  books,  and  saved  $117.  What 
was  his  salary  ? 

14.  At  Christmas  a  lady  gave  her  daughter  an  atlas  worth 

$27,  and  -f  of  the  cost  of  the  atlas  was  90%  of  the 
sum  paid  for  an  engraving.  What  was  the  sum  paid 
for  the  engraving  ? 

15.  A   sea-captain   owning  60%  of  a  vessel  gave  to  his    NL, 

son  50%  of  his  share,  which  was  worth  $6000. 
What  was  the  value  of  the  vessel  ? 

16.  A  gentleman  worth  $50,000  gave  30%  of  his  property 

to  his  son,  and  this  gift  was  80%  of  the  property 
which  the  son  already  owned.  Find  the  amount 
the  son  was  worth  after  receiving  his  father's  gift. 

205,  By  selling  a  horse  for  $90,  a  man  gains  20%  of  its 
cost.  Find  the  cost. 

He  gets  the  cost  (100%)  and  20%  of  the  cost,  or  120%  of  the  cost. 
The  question  therefore  is,  $90  is  120%  of  how  many  dollars? 

A  man  sold  a  horse  for  $90,  and  lost  25%  of  the  cost. 
What  did  the  horse  cost  ? 

He  got  the  cost  (100%)  minus  25%  of  the  cost,  or  75%  of  the  cost. 
The  question  therefore  is,  $  90  is  75  %  of  how  many  dollars  ? 

Ex.  135.     (Oral) 

1.  36  is  12-£%  more  than  what  number? 

2.  65  is  6^%  less  than  what  number? 

3.  68  is  6^%  more  than  what  number? 

4.  75  is  12^-%  less  than  what  number? 

5.  By  selling  a  hat  for  $5.40,   I  sell  it  for  20%  more 

than  the  cost.     What  was  the  cost  ? 


210  PERCENTAGE. 


6.  A  manufacturer  sells  mowing-machines  at  $  125  apiece, 

and  gains  40%.     What  do  they  cost? 

7.  Sold  a  carriage  for  $240,  which  was  20%  more  than 

the  cost.     What  was  the  cost  ? 

8.  64  is  33^%  more  than  what  number? 

9.  What  number  diminished  by  5%  of  itself  equals  190? 
10.    What  number  diminished  by  10%  of  itself  equals  180  ? 

Ex.  136. 

1.  874  is  33£%  less  than  what  number? 

2.  1740  is  20%  more  than  what  number? 

3.  40%  of  4000  is  20%  less  than  what  number? 

4.  What  number  diminished   by    15%    of  the   number 

equals  5100? 

5.  What  fraction  increased  by  25%  of  itself  equals  j-f  ? 

6.  7500  is  33£%  less  than  what  number? 

7.  A  drover  sold  250  sheep  for  $1150,  which  was  15% 

more  than  they  cost.  Find  the  cost  of  the  sheep 
per  head. 

8.  At  a  forced  sale,  a  bankrupt  sold  his  house  for  $8000, 

which  was  20%  less  than  its  real  value.  If  the 
house  had  been  sold  for  $12,000,  what  per  cent 
above  its  real  value  would  it  have  brought  ? 

9.  A  flock  of  sheep  has  been  increased  by  250%  of  its 

number,  and  now  numbers  1050.  What  is  the  origi- 
nal number? 

10.    If  20%  be  lost  on  a  ton  of  rye-straw  sold  for  $19.20, 
what  is  the  cost  of  the  straw  ? 


PERCENTAGE.  211 


PROFIT  AND  Loss. 

206.  The  difference  between  the  buying  and  selling  prices 
of  goods  is  called  profit  or  loss,  according  as  the  selling-price 
is  more  or  less  than  the  buying-price. 

Ex.  137. 

1     A  horse  which  cost  $80  was  sold  for  $60.     Find  the 

actual  loss  and  the  loss  per  cent. 
NOTE.  Gain  or  loss  is  so  much  per  cent  on  the  cost  of  the  goods. 

2.  Flour  that  cost  $10  per  barrel  was  sold  for  $12  per 

barrel.     Find  the  gain  per  cent. 

3.  If  milk  is  bought  at  4  cts.  a  quart,  and  sold  at  6  cts., 

what  is  the  gain  per  cent  ? 

4.  Goods  that  cost  $40  were  sold  at  20%   below  cost. 

What  was  the  actual  loss  ? 

5.  Velvet  is  sold  for  $3.75  per  yard,  at  a  gain  of  25%. 

Find  the  cost  of  the  velvet. 

6.  By  selling  cloth    at  $1.60  a  yard,  a  merchant  loses 

20%.     What  is  the  cost? 

7.  Five  cords  of  wood  costing  $20  were  sold  at  $7  per 

cord.     What  was  the  gain  per  cent  ? 

8.  A  carpenter  paid  $5000  for  a  house  ;  spent  in  repairs 

a  gum  equal  to  80%  of  the  purchase-price  ;  and 
then  sold  the  house  for  $12,000.  How  much  did  lie 
gain,  and  what  per  cent  of  the  whole  cost  ? 

9.  In  selling  32  yds.  of  cloth,  a  merchant  made  $6.40, 

which  was  16%  of  the  cost.  What  did  the  cloth 
cost  a  yard  ? 


212  PERCENTAGE. 


10.  Goods  were  sold  for  $1615.12£,  at  a  gain  of 

What  did  they  cost  ?  ' 

11.  If  tea  sold  at  84  cts.  a  pound  gives  a  profit  of  20%, 

what  would  be  the  profit  per  cent  if  it  were  sold  at 
75  cts.  a  pound  ? 

12.  A  trader's  profits  were  $1980  in  the  year  1880.     This 

sum  was  20%  more  than  his  profits  in  1881.     Find  • 
his  profits  in  1881. 

13.  A  cord  of  wood  costing  $4.50  sold  for  $9.     What  was 

the  gain  per  cent  ? 

14.  A  house-lot  was  sold  for  $1850,  at  an  advance  of  15% 

on  its  cost.     What  would  have  been  the  gain  per 
cent  if  it  had  been  sold  for  $2210? 

15.  A  manufacturer  owning  £  of  a  factory  sold  12^%  of 

his  share,  at  10%  above  cost,  for  $1100.     What  is 
the  cost  of  the  factory  ? 

16.  Wliat  per  cent  is  made  in  buying  coal  by  the  long  ton, 

at  $5  a  ton,  and  selling  it  by  the  short  ton,  at  the 
same  price  ? 

17.  Corn  cultivated  at  an  expense  of  28  cts.  a  bushel  is 

sold  at  1  ct.  a  pound.     What  is  the  gain  per  cent  ? 

18.  What  per  cent  advantage  is  there  in  buying  opium  by 

the  pound  avoirdupois,  and  selling  it  by  the  pound 
apothecaries'  weight? 

19.  A  grocer  lost  5%  in  selling  a  50-lb.  tub  of  butter  for 

$  15.20.     What  did  the  butter  cost  per  pound  ? 

20.  Ten  cows  were  sold  for  $690,  at  a  gain  of  15%.     For 

how  much  per  head,  on  the  average,  should,  they 
have  been  sold  to  ^ •  ji  30%  ? 


PERCENTAGE.  213 


21.  For  what  price  per  dozen  must  gloves  be  bought  in 

order  that,  by  selling  them  at  $1.75  per  pair,  there 
may  be  a  gain  of  25%  ? 

22.  A  merchant  lost  25%  by  selling  flour  at  $6  per  barrel. 

If  he  had  sold  it  at  $9  per  barrel,  what  would  have 
been  the  gain  per  cent? 

.23.  A  fruit-grower  sent  to  New  York  300  peck  baskets  of 
peaches,  valued  at  75  cts.  each.  Sixty  baskets  were 
spoiled  on  the  journey.  At  what  rate  per  basket 
must  he  sell  the  remainder  to  make  20%  profjTon 
the  entire  value  of  his  fruit  ? 

24.  Sold  goods  at  a  loss  of  20%,  and  actual  loss  of  $57.50. 

What  was  the  prime  cost  ? 

25.  Find  the  selling-price  of  goods  by  which  there  is  a  loss 

of  2%  and  an  actual  loss  of  $54.50. 

26.  How  many  pounds  of  cheese  bought  at  9  cts.  a  pound 

must  be  sold  at  12  cts.  a  pound  to  gain  $30? 

27.  Sold  steel  at  $25.44  a  ton  with  a  profit  of  6%  and  a 

total  profit  of  $  103.32.     What  quantity  was  sold  ? 

COMMISSION  AND  BROKERAGE. 

207,  The  commission  paid  to  an  agent  for  his  services  is 
generally  reckoned  at  a  rate  per  cent. 

208,  The  sum  left  after  the  payment  of  the  commission 
and  other  expenses  is  called  net  proceeds, 

209,  Commission  paid  to  a  broker  is  called  brokerage, 

210,  In  selling,  the  commission  is  reckoned  on  the  money 
received;    in   buying,   the   commission   is   reckoned   on   the 
money  paid, 


214  PERCENTAGE 


(1)  A  real-estate  agent  sold  a  house  for  $7000.     Find 
the  amount  of  his  commission,  at  \\%. 

$7000 
0.01} 

3500 
7000 

1105.00  $105.   .47W. 

(2)  A  jorkey  receives  $32  as  his  commission,  at  4%,  for 
purchasing  a  pair  of  horses.     What  did  he  pay  for  the 
horses  ? 

Commission  on  $1  invested  at  4%  is  $0.04.     Therefore  the  sum 
invested  to  obtain  a  commission  of  $32  is 

?a^  =  ?80a  $800.  Ans. 

(3)  If  a  commission  of  $212.94  is  paid  for  selling  wool 
to  the  amount  of  $6552,  what  is  the  rate  per  cent  allowed? 

^.f  the  commission  on  $6552  is  $212.94,  the  commission  on  $1  will  be 

$212^          03*. 

*  6552 
Therefore  the  commission  is  at  the  rate  of  3J%. 

(4)  A  speculator  in  New  York  sent  $  18,360  to  his  agent 
in  Chicago,  with  which  to  buy  wheat.     If  the  agent  charges 
2%  for  buying,  how  many  bushels  of  wheat  can  he  buy 
at  90  cents  a  bushel  ? 

Commission  on  $1,  at  2%,  is  $0.02.     Hence  out  of  every  $1.02 
sent,  there  is  invested  in  wheat  $  1. 

Hence,  out  of  $18,360  sent  there  is  invested  in  wheat 
$18360 

-I  Q(V)(~) 

and  the  number  of  bushels  of  wheat  bought  is  =  20,000. 

20,000.  Ans. 


PERCENTAGE  215 


Ex.  138. 

1.  A  commission-merchant  sold  90  bbls.  of  flour  at  $6  a 

barrel,  and  received  5%  commission.  What  was  his 
commission  ? 

2.  A  commission  of  $  121.29  was  charged  for  selling  $1866 

worth  of  goods.     What  was  the  rate  of  commission  ? 

3.  A  grain-dealer  charged  7%  commission  for  selling  a 

quantity  of  wheat,  and  received  for  his  commission 
$109.20.  What  was  the  total  amount  received  for 
the  wheat? 


4.  A  real-estate  broker  sold  a  house  on  §\%  commission, 

and  sent  to  the  owner  as  net  proceeds  $3060.  What 
was  the  broker's  commission,  and  what  sum  was 
received  for  the  house? 

HINT.  The  broker  received  f>]%,  and  the  owner  93^%,  of  the 
sum  the  house  sold  for.  Hence  the  question  is,  $3060  is  933% 
of  what  sum  ? 

5.  A  New  York  merchant  sent  $1295.32  to  New  Orleans 

to   be    expended   in   cotton.     The   broker  in   New 

Orleans  charged  6%  commission.     What  sum  was 

paid  for  cotton  ? 

HINT.  The  broker  received  C%  commission  on  the  money 
invested  in  cotton.  Therefore,  the  question  is:  $1295.32  is 
106%  of  what  sum? 

6.  If  $5125  include  the  amount  expended  for  wool  and 

2^%  commission  to  the  purchasing  agent,  how  much 
money  does  the  agent  lay  out  in  wool  ? 

7.  A  lawyer  collected  75%  of  a  debt  of  $1260,  and  charged 

5%  commission  on  the  sum  collected.  What  did  the 
creditor  receive? 


216  PERCENTAGE. 


8.  An  agent  sold  420  bu.  of  corn  at  60  cts.  a  bushel,  and 

the   commission    was   $7.56.     What   rate   of  com- 
mission was  charged  for  selling  ? 

9.  A  land  agent  charged  4%  for  selling  750  A.  of  land 

at  $  20  an  acre.     What  was  his  commission  ? 

10.  How  many  yards  of  cloth,  at  45  cts.  a  yard,  can  an 
agent  buy  with  the  commission  received  from  the 
sale  of  180  bu.  of  potates  at  50  cts.  a  bushel,  his  rate 
of  commission  for  selling  the  potatoes  being 


11.  A  man  bought  a  horse  for  $225,  which  sum  was  half 

of  his  commission,  at  %%%,  on  the  sale  of  a  farm. 
What  did  the  farm  bring  ? 

12.  A  young  man  selling  tea  on  2|%  commission  sent  to 

his  employer  $875.25  as  the  net  proceeds  of  one 
week's  sales.  What  did  the  average  daily  sales 
amount  to  ? 

13.  A  St.  Louis  merchant  received  $150  as  his  commission, 

at  2J%  for  purchasing  1200  bbls.  of  flour.  What 
was  the  price  paid  per  barrel  ? 

14.  A  broker  sold  for  a  farmer  12,000  Ibs.  of  pork,  at  8| 

cts.  per  pound.  He  charged  3%  commission  for 
selling,  and  paid  $37.60  for  freight.  How  many 
feet  of  pine  boards,  at  $25  per  M.,  can  the  broker 
buy  with  the  net  proceeds,  if  he  charges  1%  com- 
mission for  buying  ? 

15.  A  broker  is  offered  a  commission  of  5^%  for  selling 

wool  and  guaranteeing  payment,  or  a  commission  of 
3£%  without  guaranteeing  payment.  He  accepts 
the  $>\°/o  commission,  and  guarantees  the  payment. 
The  sales  amount  to  $8500,  and  the  bad  debts  to 
$147.75.  How  much  did  he  gain  by  his  choice  ? 


PERCENTAGE.  217 


INSURANCE. 

211.  In  insurance  a  payment  called  a  premium  of-  insur- 
ance is  made  for  a  guaranty  of  a  specified  sum  of  money 
in  the  event  of  loss  from  fire  or  accident,  and  is  reckoned 
at  a  rate  per  cent  on  the  amount  insured. 

212.  In    life-insurance  an  annual  payment  is  made  in 
order  to  secure  a  specified  sum  of  money  in  the  event  of 
death,  or  at  the  end  of  a  fixed  period  of  time. 

213.  The  written  contract  is  called  the  policy  of  insurance, 

(1)  A  house  worth  $8000  is  insured  for  three  years  for  f 
of  its  value,  at  1%.     Find  the  premium. 

J  of  $  8000  -  $  6000,  and  1%  of  $  6000  =  $  60.     $  60.   Ans. 


(2)  The  premium  for  insurance  on  a  store,  at  1^%,  is 
$150.     Find  the  amount  of  the  insurance. 


The  premium  on  $1  insurance,  at  1J%,  is  $0.015. 

Of 
$10,000.  Ans. 


150 

Hence  the  amount  of  insurance  is  $ — —  =  $10,000. 

0.015 


(3)  A  man  pays  $27.50  premium  for  having  his  house 
insured  for  five  years,  at  \\°/o  on  \  of  its  value.  What  is 
the  value  of  the  house  ? 


The  premium  on  $1  insured  at  1J%  is  $0.0125. 

9*7   CA 

Hence  the  amount  of  the  insurance  is  $-^— '• —  =  $2200,  and  the 
value  of  the  house  is  $  2200  -*-  §  =  $  3300.  $  3300.    Ans. 


218  PERCENTAGE. 


(4)  For  what  sum  must  a  cargo  worth  $24,500  be  in- 
sured, at  2%,  so  that,  in  case  of  loss,  the  owner  may  recover 
both  the  value  of  the  cargo  and  the  premium  paid  ? 

Premium  on  $1  at  2%  is  $0.02. 

Insurance  on  $0.98  worth  of  cargo  =  $1. 

94.^00 

Hence  insurance  on  $24,500  worth  of  cargo  =  $  -     -  =  $  25,000. 

u.yo 

$25,000.  Ans. 
Ex.  139. 

1.  Find  the  cost  of  insuring  property  worth  $15,000,  if  f 

of  the  value  is  insured  at  £$>. 

2.  Find  the  cost  of  insuring  |-  of  the  value  of  6000  bbls. 

of  flour  worth  $9.60  a  barrel,  the  insurance  being 
reckoned  at  \°Jo. 

3.  A  stock  of  goods  worth  $12,000  was  insured  for  \  of 

its  value  at  f  %.  If  the  whole  stock  were  burned, 
what  would  be  uie  loss  to  the  owner,  including  the 
premium  paid  for  insurance? 

4.  After  three  annual  payments  of  $337.50,  premium  at 

\\°fo  on  f  of  the  value  of  a  mill,  it  was  burned. 
Find  the  loss  to  the  insurance  company. 

5.  At  \°/o,  how  much  insurance  can  be  effected  upon  a 

store  for  $108? 

6.  What  annual  premium  at  \\°/o  must  be  paid  on  a  life- 

insurance  of  $6000? 

7.  At  the  rate  of  $  17  upon  $  1000,  what  annual  premium 

will  be  paid  on  a  life-insurance  of  $6700  ? 

8.  The  annual  premium  paid  for  life-insurance  at  If  % 

is  $70.     What  is  the  sum  insured  ? 

9.  For  what  sum  should  a  cargo  worth  $74,496  be  in- 

4sured,  at  3%,  so  that,  in  case  of  loss,  the  owner  may 
recover  both  the  value  of  the  cargo  and  the  premium 
paid? 


PERCENTAGE.  219 


TAXES  AND  DUTIES. 

214,  Taxes  on  property  are  reckoned  at  a  rate  per  cent 
on  the  assessed  value  of  the  property ;  and  duties  on  im- 
ported goods  are  sometimes  reckoned  at  a  rate  per  cent  on 
the  cost  in  the  country  from  which  they  are  imported. 

Ex.  A  tax  of  $18,000  is  levied  upon  a  town  which  contains 
800  polls,  assessed  at  $1.50  each,  and  which  has 
taxable  property  valued  at  $1,100,000.  It  is  esti- 
mated that  the  town  will  receive  from  the  state 
$3600  as  its  share  of  the  railroad  tax.  Find  the 
rate  of  taxation  and  the  tax  paid  by  Brown,  whose 
property  is  assessed  at  $5960,  and  who  pays  for  1 
poll. 

The  amount  of  poll-taxes  =  800  X  $1.50  =  $1200 

The  amount  from  the  state  =  $3600 

The  sum  from  state  and  polls  =  $4800 

Sum  levied  on  property  =  $18,000  -  $4800  =  $13,200. 

The  rate  =  $13,200  -4-  $1,100,000  =  $0.012. 

That  is,  the  tax  is  12  mills  on  a  dollar,  or  $12  on  $1000. 

Therefore  Brown's  property-tax  is  0.012  of  $5960  =  $71.52. 

Total  tax  -  $71.52  +$1.50  =  $73.02. 

Ex.  14O. 

1.  If  the  assessed  valuation  of  a  town  is  $784,750,  and 

the  town  has  260  polls,  paying  $1.25  each,  what  is 
the  rate  when  the  tax  levy  is  $16,020  besides  the 
estimated  amount  to  be  received  from  the  state  ? 

2.  A  district  schoolhouse  is  to  cost  $3500,  and  the  prop- 

erty of  the  district  is  assessed  at  $210,000.  What 
is  the  rate,  and  what  tax  must  be  paid  on  property 
assessed  at  $3798.60? 


220  PERCENTAGE. 


3.  In  a  city  of  2000  polls,  each  paying  $1.50,  the  sum  of 

$111,000  is  to  be  raised  by  taxation  on  property 
assessed  at  $9,000,000.  What  is  the  tax  of  a  man 
who  pays  for  4  polls,  and  tax  on  property  assessed 
at  $25,670? 

4.  What  is  the  rate  of  taxation  when  $710.92  is  the  tax 

upon  $50,780? 

5.  If  a  tax  of  $12,350  is  to  be  raised,  and  the  collector 

receives  5%  for  collecting  the  taxes,  what  sum  must 
be  levied? 

6.  A  town-hall  is  to  be  built  at  a  cost  of  $  11,400.    What 

sum  must  be  assessed  if  the  collector  receives  5% 
for  collecting  the  taxes,  and  what  will  be  the  rate 
if  the  assessed  valuation  of  the  town  is  $800,000? 

7.  Find  the  duty,  at  15%,  on  95  cases  of  indigo,  each 

weighing  190  Ibs.,  and  invoiced  at  75  cts.  per  pound. 

8.  After  deducting  20%  for  leakage,  what  will  be  the 

duty  on  40  hhds.  of  molasses,  of  84  gals,  each,  if  the 
molasses  is  invoiced  at  90  cts.  a  gallon,  and  the  duty 
is  30%  ? 

9.  On  15  doz.  bottles  of  sherry  wine  there  is  paid  $1.25 

per  dozen  for  transportation,  and  $1.50  per  dozen 
for  duty.  What  is  the  whole  cost  of  importation  ? 

10.    A  Boston  merchant  received  from  Paris  : 
325  yds.  of  silk       @  $2.25  a  yard ; 
296  yds.  of  lace      @      1.50  a  yard ; 
480  yds.  of  ribbon  @      0.50  a  yard ; 

45  doz.  gloves       @    15.00  a  doz. 
If  the  duty  en  silk,  ribbon,  and  lace  is  35%,  and  on 
gloves   25%,   what   is   the   whole   amount   of    the 
duties  9 


PERCENTAGE.  221 


11.  If  the  duty  is  $2.50  a  gallon  on  cologne- water,  what 

must  be  paid  on  75  doz.  pint  bottles,  if  there  is  an 
allowance  of  5%  for  breakage? 

12.  What  is  the  invoice  cost  of  goods  upon  which  $625 

duty  is  paid,  if  the  duty  is  reckoned  at  25%  ? 

13.  What  will  be  paid  by  a  grocer  importing  120  chests 

of  tea,  containing  79  Ibs.  each,  invoiced  at  75  cts. 
per  pound,  if  the  duty  is 


Ex.  141. 
MISCELLANEOUS  EXAMPLES. 

1.  Of  what  number  is  450  nine  per  cent? 

2.  What  is  the  excess  of  5%  of  1500  over  \%  of  7000  ? 

3.  What  per  cent  of  9000  is  45? 

4.  Five   hundred   and   sixty   is   12%    more   than   what 

number  ? 

5.  Seven  hundred  and  fifty-two  is  6%  less  than  what 

number  ? 

6.  There  is  a  difference  of  893  between  a  certain  number 

and  6%  of  the  number.     Find  the  number. 

7.  What  per  cent  of  25  Ibs.  are  3  Ibs.  4  oz.  ? 

8.  The  difference  between  50£%  and  75J%  of  a  number 

is  99.      Find  the  number.      Let  the   example   be 
proved. 

9.  A  merchant  sold  cloth  at  $4.20  per  yard,  and  gained 

20%.     If  it  had  been  sold  at  $3.60,  what  actual 
gain,  and  what  gain  per  cent,  would  have  been  made  ? 

10.    By  how  much  does  £%  exceed  £%  ? 


222  PERCENTAGE. 


11.  At  an  average  price  of  55  cts.  per  bushel,  and  a  charge 

of  2^-%  commission,  how  many  bushels  of  grain  can 

be  bought  for  $4510? 

HINT.    First  find  the  cost  of  1  bu.f  including  commission. 

12.  A   landau  was   sold   for  $488,   at   a   gain   of  22%. 

Kequired  the  cost. 

13.  A  milkman's  gallon  measure  was  too  small  by  ^  gi. 

What  was  the  rate  per  cent  of  fraudulent  gain  ? 

14.  A  merchant  paid  $112.50  for  75  yds.  of  silk,  of  which 

15  yds.  were  worthless.  At  what  price  per  yard 
must  the  remainder  be  sold  to  gain  20%  on  the 
purchase-price  of  the  whole  ? 

15.  For  selling  goods,  an  agent  received  $106.83  commis- 

sion, 2J%  for  selling,  2f%  for  guaranteeing  pay- 
ment. What  sum  was  received  for  the  goods? 

16.  A  dealer  bought  70  bags  of  wool  at  $32  a  bag ;  10% 

of  it  proved  unsalable.  For  what  price  per  bag 
must  he  sell  the  rest  to  realize  15%  on  his  pur- 
chase ? 

17.  A  lady  paid  for  investing  money  $9.37|  brokerage, 

rate  |%.     Required  the  amount  invested. 

18.  From  a  stack  of  hay,  7  t.  11  cwt.  were  sold,  which  was 

75£%  Of  the  whole.  What  did  the  stack  contain 
before  the  sale  ? 

19.  A  carriage  worth  $250  was  bought  for  $50  less,  and 

sold  for  $25  more,  than  its  value.  What  was  the 
rate  of  gain  on  the  price  paid  ? 

20.  A  man  left  30%  of  his  estate  to  his  wife,  50%  of  the 

remainder  to  his  son,  75%  of  the  residue  to  his 
daughter,  and  the  balance,  $546,  to  a  family  servant. 
Required  the  value  of  the  estate. 


PERCENTAGE.  223 


21.  What  per  cent  of  ^  is  ^  ?  of  T7¥  is  $*  ? 

22.  A  man  sold  36  horses  for  $200  each :  on  half  of  them 

he  gained  20%,  and  on  half  he  lost  10%.  What 
was  his  gain  per  cent  on  the  whole  sale  ? 

23.  A  gentleman  sent  to  a  broker  $1281.25  to  be  invested 

in  land  at  $62.50  an  acre.  A  commission  of  2^-% 
being  charged  for  buying,  how  many  acres  were 
bought? 

24.  The  dimensions  10,  8,  and  6,  of  a  rectangular  bin  being 

increased  10%,  what  will  be  the  rate  per  cent  of 
increase  in  capacity? 

25.  One-half  of  a  stock  of  goods  valued  at  $612.60  was 

sold  for  •§-  of  the  value  of  the  whole  stock.  What 
was  the  gain  per  cent  ? 

26.  A  roll  of  140  yds.  of  carpet  was  sold  for  $72,  at  a  loss 

of  10%.  What  should  it  have  brought  per  yard  to 
insure  a  gain  of  15%  ? 

27.  A  railroad  company  with  $9,000,000  capital  declares 

a  dividend  of  $360,000.  What  sum  will  be  received 
on  120  shares  of  $100  each? 

28.  Ten  per  cent  of  a  roll  of  carpet  having  been  sold  to 

one  man,  10%  of  the  remainder  to  another,  30.375 
yds.  are  left.  How  many  yards  were  there  at  first? 

29.  At  an  annual  premium  of  $405,  rate  1$%,  f  of  the 

value  of  a  mill  is  insured.  What  is  the  entire  value 
of  the  mill  ? 

30.  A  broker  buying  cotton  at  f  %  commission  retained 

$75  for  his  commission,  and  paid  $25  for  storage. 
What  sum  was  sent  by  his  employers  to  cover  the 
whole  expense  of  investment  ? 


224  PERCENTAGE. 


31.  What  sum  must  be  insured  upon  a  library  to  cover  its 

entire  value,  $18,000,  and  the  premium  at  If  %  ? 

HINT.  If  100  be  taken  to  represent  the  sum  to  ba  insured, 
then  If  will  represent  the  premium  ;  and  100 -If,  that  is,  98  J, 
will  represent  the  value  of  the  library.  Hence  the  sum  to  be 
insured  will  be  $  18,000  H-  0.98$  =  f  18,329.94. 

32.  A  merchant  placed  80%  of  his  year's  profits  in  a  bank ; 

having  drawn  out  20%  of  this  deposit,  $2880 
remained.  What  were  his  profits  for  the  year  ? 

33.  Required  the  tax-rate,  in  a  city  appropriating  for  pub- 

lic expenses  $147,000,  to  be  assessed  on  property 
worth  $35,000,000. 

34.  A  lady  bought  a  house  for  $7965,  which  rented  for 

$841.85.  The  taxes  were  $50;  repairs,  $75.  What 
rate  per  cent  did  the  investment  yield  ? 

35.  A  premium  of  $960  was  paid  for  full  insurance  on  a 

ship  and  cargo,  at  1£%.  The  cost  of  the  cargo  was 
60%  of  the  cost  of  the  ship.  What  was  the  value 
of  each  ? 

36.  Find  the  entire  cost  of  4000  bbls.  of  flour  purchased 

by  an  agent,  at  $7  a  barrel,  who  charged  3%  com- 
mission, and  paid  $315  for  freight. 

37.  How  many  barrels  of  flour  can  be  bought  for  $5924,38 

by  an  agent  who  pays  $7  a  barrel  for  the  flour, 
charges  3%  commission,  and  pays  $315  for  the 
freight? 

38.  The  insurance  on  -|  the  value  of  a  hotel  and  furniture 

cost  $300.  The  rate  being  75  cts.  on  $100,  what 
was  the  value  of  the  property  ? 

39.  What  is  the  duty,  at  25f  cts.  per  gallon,  on  48  bbls. 

of  turpentine,  31  gals,  making  a  barrel,  and  b% 
being  allowed  for  leakage  ? 


CHAPTER  XL 

INTEREST  AND   DISCOUNT. 

215.  Interest  is  the  payment  made  for  the  use  of  money. 
The  interest  to  be  paid  for  the  use  of  a  given  sum  of  money  differs 

from  the  payments  considered  in  the  last  chapter,  in  that  it  depends 
upon  the  time  for  which  the  sum  is  loaned  as  well  as  on  the  rate  per 
cent  charged. 

216.  The  sum  loaned  is  called  the  principal.     The  princi- 
pal and  interest  together  is  called  the  amount. 

SIMPLE  INTEREST. 

217.  If  100  be  taken  as  the  representative  of  the  princi- 
pal, the  rate  will  represent  the  interest  for  one  year;  the 
product  of  the  rate  by  the  number  of  years  will  represent 
the  whole  interest. 

Thus,  if  the  time  be  4  yrs.,  and  the  rate  per  cent  5,  the  interest  will 
be  represented  by  20,  and  the  amount  by  120. 

Find  the  interest  on  $512  for  2  yrs.  4  mos.,  at  6%. 

$512 
0.06 

$30.72    -  interest  for  1  yr. 
2-^  =  2  yrs.  4  mos. 

1024 
6144 
$71.68  $71.68.  Am. 

218.  In  most  business  transactions  the  time  for  which 
interest  is  required  is  1,  2,  3,  or  4  months  (30  dys.  being 


226  INTEREST  AND  DISCOUNT. 

reckoned  1  mo.),  and  the  rate  of  interest  is  6%,  that  is, 
\°}0  a  month. 

Hence  the  interest  at  6%  on  a  given  sum  for  2  mos.  (or 
60  dys.)  is  found  by  moving  the  decimal-point  two  places 
to  the  left;  for  1  mo.,  3  mos.,  4  mos.,  by  moving  the  deci- 
mal-point two  places  to  the  left,  and  multiplying  by  \,  \\, 
and  2  respectively. 

Thus,  the  interest  on  $2500  for  2  mos.  is  $25.00 ;  for  1  mo.,  $  12.50 ; 
for  3  mos.,  $37.50  ;  for  4  mos.,  $50 

Find  the  interest  on  $1120  for  3  yrs.  2  mos.  18  dys., 
at  6%. 

The  interest  at  6%  for  1  yr.  -  0.06  of  the  principal. 
The  interest  for  1  mo.  is  ^  of  0.06  =•  0.005  of  tho  principal. 
The  interest  for  1  dy.  is  ^  of  0.005  -  i  of  0.001  of  the  principal. 
Hence  the  interest  for  . 

3  yrs.   =3x0.06  =0.18 

2  mos.  =  2  x  0.005  =  0.01 
18  dys.  =  18  X  J  of  0.001  =  0.003 

3  yrs  2  mos.  18  dys.          -  0.193  of  the  principal. 
And  0.193  of  $1120  =  $216.16. 

$216.16.  Ans. 

219.  The  six  per  cent  method  may  be  employed  for  any 
rate  per  cent  by  first  finding  the  interest  at  6%,  and  then 
taking  such  a  part  of  the  interest  as  the  given  rate  is  of  six 
per  cent. 

Thus,  the  interest  at  4£  %  =  ^  «=  f  of  the  interest  at  6  %.     In  this 

6 

case,  we  should  diminish  the  interest  at  6%  by  J  of  itself.  The  inter- 
est at  8%  is  -|  =  f  of  the  interest  at  6%.  In  this  case,  we  increase 
the  interest  at  6%  by  J  of  itself. 

220,  To  compute  interest  for  days  at  6%,  we  move  the 
decimal-point  in  the  principal  three  places  to  the  left,  and 
multiply  by  one-sixth  of  the  number  of  days. 


INTEREST  AND  DISCOUNT.  227 

Find  the  interest  for  $8080  for  93  Ays.,  at  6%. 

$3.08  By  moving  the  decimal-point  three  places  to  the 

151  left,  we  have  $8.08  ;  and  £  of  93  dys.  =  15  J.    There- 

404  fore,   multiplying  $8.08    by   15£,   we    obtain   the 

4040  required  interest. 

$125.24.  Ans. 


221.  For  any  other  rate,  find  the  interest 
at  6%,  and  then  increase  or  diminish  this  interest  by  such 
a  fraction  of  itself  as  the  given  rate  is  greater  or  less 
than  6%. 

Ex.  142. 

Find  the  interest  of : 

1.  $51.25  for  30  dys.,  at  6% 

2.  $2581  for  60  dys.,  at  6%. 

3.  $1261  for  90  dys,  at  6%. 

4.  $  1250.60  for  4  mos,  at  6%. 

5.  $3020  for  3  mos.,  at  6%. 

6.  $2300  for  3  mos,  at  6%.  ^ 

7.  $275  for  2  mos,  at  6%. 

8.  $5000  for  1  mo,  at  6%. 

9.  $1361  for  2yrs,  at  5%. 

10.  $675.90  for  5  yrs,  at  3|-%. 

11.  $775.83  for  3  yrs.  9  mos,  at 

12.  $533.33^  for  10  mos,  at  4|$> 

13.  $250.60  for  3  yrs.  6  mos,  at 

14.  $575.87|  for  1  yr.  10  mos.  15  dys,  at  5%. 

15.  $760  for  2  yrs.  11  mos.  27  dys,  at  ±\%. 


228  INTEREST  AND  DISCOUNT 


16.  $725.40  for  5  mos.  27  Ays.,  at 

1.7.  $547.60  from  Feb.  20  to  Dec.  5,  at 

18.  $1750  from  May  5,  1884,  to  June  21,  1885,  at 

19.  $1517  from  Jan.  5  to  July  1,  at  4|%. 

20.  $476.50  from  July  5,  1884,  to  Feb.  9,  1885,  at 

21.  $319.20  from  April  7  to  Aug.  31,  at  3J%. 

22.  $6460  from  June  15,  1883,  to  May  7,  1885,  at 

23.  $150  from  Aug.  5,  1883,  to  March  17,  1885,  at 

24.  $527.20  from  Jan.  1  to  Nov.  20,  at  4|%. 

25.  $1250  from  Nov.  15,  1884,  to  March  1,  1885,  at 

26.  $624.36  from  March  5  to  Dec.  20,  at 


Find  the  amount  of  : 

27.  $  1100  for  3  yrs.  4  mos.,  at  5%. 

28.  $1290.50  for  60  dys.,  at  6%. 

29.  $1275  for  3  yrs.  2  mos.  15  dys.,  at  8%. 

30.  $250.80  for  10  mos.  10  dys.,  at 

31.  $377.65  for  1  yr.  3  mos.,  at  5 

32.  $7234.25  for  22  yrs.  2  mos.  20  dys.,  at 

33.  $6130  from  May  6  to  Oct.  24,  at  3}%. 

34.  $258.85  from  March  6  to  June  24,  at  5%. 

35.  $25.62  for  33  dys,  at  6%. 

36.  $85.85  for  1  yr.  7  mos.  21  dys,  at  6%. 

37.  $600  for  93  dys,  at  4%. 

38.  $350  from  Sept.  21,  1884,  to  March  b,  1885,  at 


INTEREST  AND  DISCOUNT.  229 

39.  $1226  from  Oct.  4,  1884,  to  May  6,  1885,  at  5%. 

40.  $342.42  from  Feb.  5,  1884,  to  March  15,  1885,  at  7%. 

41.  $360.50  from  Aug.  1, 1884,  to  March  3,  1885,  at  6|%. 

42.  $504.25  from  Jan.  8  to  March  10,  at  6J%. 

43.  $1240  from  Mar.  3  to  Aug.  28,  at  7%. 

NOTE.  In  business,  a  year  is  reckoned  at  360  days  in  computing 
interest  for  a  time  less  than  a  year  expressed  in  months  and  days ; 
hence  the  interest  is  y£^  or  ^  too  great.  But  national  governments 
take  the  number  of  days  between  the  two  given  dates,  and  reckon  for 
the  interest  such  a  part  of  a  year's  interest  as  this  number  of  days  is 
of  365  days. 

222.  It  is  often  required  to  find  the  rate,  time,  or  prin- 
cipal, when  two  of  these  and  the  interest  (or  amount)  are 
given. 

223.  When  the  principal,  interest  (or  amount),  and  time 
are  given,  to  find  the  rate  per  cent. 

At  what  rate  per  cent  will  $320  produce  $48  in  3  yrs.? 
Interest  on  $320  for  3  yrs.  is  $48. 
Interest  on  $320  for  1  yr.  is  J  of  $48. 
Interest  on  $  1  for  1  yr.  is  ^  of  J  of  $48  =  $0.05. 
But  $0.05-  5%  of  fl.  5^   Am 

At  what  rate  per  cent  will  $8000  amount  to  $9277.78 
in  2  yrs.  6  mos.  20  dys.  ? 

Interest  is  $9277.78  -  $8000  =  $1277.78. 
Time  is  2  yrs.  6  mos.  20  dys.  =  2f  yrs. 
'  Interest  on  $8000  for  2f  yrs.  =  $1277.78. 
Interest  on  $8000  for  1  yr.     -  f  1277.7a 

Interest  on  $  1  for  1  yr.  - 1^§-  $°-06l- 

But  $0.06i  -6J%  of  |1. 


230  INTEREST  AND  DISCOUNT. 

Ex.  143. 
Find  the  rate  per  cent  : 

1.  When  the  interest  on  $500  for  1  yr.  6  mos.  is  $67.50. 

2.  When  the  interest  on  $250  for  2  yrs.  is  $52.50. 

3.  When  $500  amount  to  $754  in  9  yrs. 

4.  When  the  interest  on  $725  for  12  yrs.  is  $141.37f 

5.  When  $880  amount  to  $899.25  for  7  mos. 

6.  When  the  interest  on  $424  for  2  yrs.  6  mos.  is  $26.50. 

7.  When  the  interest  on  $255.50  from  April  1  to  June 

20  is  $2.80. 

8.  When  $175  amount  to  $203.35  for  3  yrs.  7  mos.  6  dys. 

9.  When  a  sum  of  money  is  doubled  in  16  yrs. 

10.    When  an  investment  for  6  yrs.  produces  a  sum  equal 
to  •§•  of  the  capital. 

224,    When  the  principal,  interest  (or  amount),  and  rate 
per  cent  are  given,  to  find  the  time. 

In  what  time  will  the  interest  on  $793.87£  be  $11.96£, 


Interest  on  $793.875  at  5J%  for  1  yr.  -  $43.663. 
Therefore  the  number  of  yea 
And  0.274  yr.  =  3  mos.  9  dys. 


Therefore  the  number  of  years  will  be  11-965  =  0.274. 

43.663 


3  mos.  9  dys.  Ans 


Ex.  144. 
Find  the  time  in  which  : 

1.  The  interest  on  $225  will  be  $36,  at  4%. 

2.  $440  will  amount  to  $505.45.  at 


INTEREST  AND  DISCOUNT.  231 


3.  $2ioO  will  double  itself,  at 

4.  $225  will  amount  to  $256.50,  at 

5.  $50  will  amount  to  $85,  at  6%. 

6.  The  interest  on  $4260  will  be  $873.30,  at 

7.  $1005.34  will  amount  to  $1156.14,  at 

8.  $1587.75  will  amount  to  $1611.68,  at 

9.  A  sum  of  money  will  double  itself,  at  6%. 
10.  $1000  will  amount  to  $1125,  at  4%. 

225,   When  the  interest,  time,  and  rate  are  given,  to  find 
the  principal. 

What  principal  will  in  8  yrs.  6  mos.  produce  $100  in- 
terest, at  5%? 

8  yrs.  G  mos.  =  8.5  yrs. 
Interest  for  1  yr.  =  $199  =  $11.705. 
Interest  on  $1  for  1  yr.  at  5%  =  0.05  of  $1. 

>3I 

$235.30.  Ans. 


Hence  principal  required  -  ^1L765  «=  $235.30 
0.05 


Ex.  145. 

Find  the  principal  that  will : 

1.  Produce  $180  interest  in  3  yrs.,  at 

2.  Produce  $189  interest  in  3  yrs.,  at 

3.  Produce  $3493.20  interest  in  3  yrs.  5  mos.,  at 

4.  Produce  $10.70  interest  in  5  mos.,  at  4%. 

5.  Produce  $75.40  interest  in  3  yrs.  4  mos.,  at 

6.  Produce  $75.05  interest  in  3  mos.  2  dys.,  at 


232  INTEREST  AND  DISCOUNT. 

7.  Produce  $1746.60  interest  in  3  yrs.  5  mos.,  at  (j%. 

8.  Produce  $64.46  interest  in  6  yrs.,  at  4£%. 

9.  Produce  $80.62£  interest  in  3  yrs.  9  mos.,  at  4%. 

10.    Produce  $669.64  interest  in  2  yrs.  7  mos.  24  dys., 
at  6%. 

226,   When  the  amount,  time,  and  rate  are  given,  to  find 
the  principal. 

Find  the  principal  that  will  amount  to  $748.12|  in  3  yrs. 
6  raos.,  at  4%. 

3  yrs.  6  mos.  =  3J  yrs. 

Let  the  principal  be  represented  by  100. 

The  interest  will  be  represented  by  3  J  X  4  =  14. 

The  amount  will  be  represented  by  100  +  14  =  114. 

Hence  the  principal  =  fJJ  of  $748.125  =  $656.25. 

$656.25.  Ans, 


Ex.  146. 

Find  the  principal  that  will  amount  : 

1.  To  $1680  in  3  yrs.,  at  4%. 

2.  To  $962  in  4£  yrs.,  at  4£%. 

3.  To  $725.47  in  2  yrs.  3  mos.,  at  3|%. 

4.  To  $3215.83  in  4  yrs.  6  mos.,  at  3%. 

5.  To  $595.20  in  8  mos.,  at  6%. 

6.  To  $1275.75  in  1  yr.  1  mo.,  at  5%. 

7.  To  $2053.  32  in  3  yrs.  5  mos.,  at  6%. 

8.  To  $131.88  in  2  yrs.  11  mos.  15  dys.,  at 

9.  To  $37.02  in  2  yrs.  3  mos.  18  dys.,  at  5 
10.  To  $2359.38  in  2  yrs.  7  mos.  24  dys.,  at 


INTEREST  AND   DISCOUNT.  233 

BANK  DISCOUNT. 

227,  When  the  holder  of  a  promissory  note  sells  the 
note  to  a  bank,  or  other  purchaser,  the  sum  paid  by  the 
bank   is   called   the   proceeds   or  avails   of  the    note,  and 
the  difference  between  the  sum  named  in  the  note  and  the 
proceeds  is  called  the  discount, 

228,  Discount  is  reckoned  at  so  much  per  cent,  and  the 
per  cent  is  called  the  rate  of  discount, 

229,  Questions  in  bank  discount  are  calculated  like  ques- 
tions in  simple  interest,  the  terms   used   being   discount 
instead  of  interest,  and  rate  of  discount  instead  of  rate  of 
interest. 

NOTE.  The  sura  named  in  the  note  should  he  written  in  words,  and 
is  called  the  face  of  the  note.  The  person  signing  the  note  is  called 
the  maker ;  a  person  who  writes  his  name  on  the  back  of  the  note  is 
called  an  indorser,  and  is  responsible  for  the  payment  of  the  note. 

A  note  should  contain  the  words  "  value  received.1'  A  note,  to  be 
negotiable,  must  be  made  payable  to  the  bearer,  or  to  the  order  of 
some  person  who  must  indorse  the  note. 

When  a  note  bears  interest,  the  discount  is  computed  on  the  amount 
of  the  note. 

A  note  is  nominally  due  at  the  expiration  of  the  time  named  in 
the  note,  but  it  does  not  mature,  that  is,  become  legally  due,  until 
three  days  after  this  time.  These  three  days  are  called  days  of 
grace.  And  the  discount  is  computed  on  the  time  between  the  day 
the  note  is  discounted  and  the  day  of  its  maturity. 

When  the  time  is  expressed  in  days,  the  day  of  maturity  is  found 
by  counting  forward  from  the  date  of  the  note  the  number  of  days 
named  in  the  note,  and  the  three  days  of  grace.  When  the  time  is 
in  months,  the  day  of  maturity  is  found  by  counting  the  number  of 
calendar  months,  and  the  three  days  of  grace.  When  a  note  falls  due 
on  Sunday,  or  a  legal  holiday,  it  is  payable  on  the  day  previous. 

A  protest  is  a  notice  in  writing  by  a  notary  public  to  the  indorsers 
that  a  note  has  not  been  paid.  If  a  note  be  not  protested  on  the 
last  day  of  grace  the  indorsers  are  released  from  their  obligation. 


234  INTEREST  AND  DISCOUNT. 

230,  Find  the  day  of  maturity,  the  time  to  run  (from  the 
day  the  note  is  discounted),  the  discount,  and  the  proceeds 
of  the  following  notes  : 

$610.25.  BOSTON,  June  12,  1885. 

Sixty  days  after  date  I  promise  to  pay  to  the  order  ol 
Edwin  Ginn  six  hundred  ten  and  -j2^  dollars,  for  value 
received. 

Discounted  at  6%,  July  1.  SAMUEL  HALE. 

Counting  60  dys.  from  June  12,  we  have  18  in  June,  31  in  July, 
and  11  in  August. 

Therefore  the  note  becomes  due  Aug.  n/u  (11  denotes  the  day  it 
is  nominally  due,  and  14  the  day  it  is  legally  due). 

The  time  to  run  is  30  dys.  in  July  and  14  in  August,  that  is, 
44  dys. 

The  discount  is  the  interest  on  $610.25  for  44  dys.,  at  6%.  There- 
fore (J  220)  the  discount  is  7£  X  $0.61025  =  $4.48. 

T^e  proceeds  -  $  610.25  -  $4.48  =  $605.77. 

Due  Aug.  14;  discount,  $4.48;  proceeds,  $605.77.  Ans. 


$  1050.  CHICAGO,  Feb.  13, 1885. 

Six  months  from  date  we  jointly  and  severally  promise 
to  pay  to  the  order  of  George  Hall  ten  hundred  and  fifty 
dollars,  for  value  received,  with  interest  at  six  per  cent. 

Discounted  at  8%,  May  13.  JAMES  BLAKE. 

HENRY  SHAW. 

interest  on  note  for  6  mos.  3  dys.  =  $32.03. 

Amount  of  note  when  due  is  $  1050  +  $32.03  =  $  1082.03. 

Day  of  maturity,  Aug.  13/16. 

Time  to  run,  3  mos.  3  dys. 

Discount  on  $1082.03,  at  8%,  for  3  mos.  3  dys.  =  $22.36. 

Proceeds,  $  1082.03  -  $  22.36  =  $  1059.67. 

Due  Aug.  16;  discount,  $22.36 ;  proceeds,  $1059.67.  An* 


INTEREST  AND  DISCOUNT.  235 


Ex.  147. 

Find  the  day  of  maturity,  the  time  to  run,  the  discount, 
and  the  proceeds,  on  the  following  notes  : 

1.  $2250.  CONCORD,  N.H.,  Jan.  1,  1885. 
Four  months  from  date  I  promise  to  pay  to  the  order  of 

George  Marston  twenty-two  hundred  and  fifty  dollars,  for 
value  received. 

Discounted  at  1%,  Jan.  12.  SIMON  STEVENS. 

2.  $432.55.  NEW  YORK,  Jan.  3,  1885. 
Sixty  days  from  date  I  promise  to  pay  James  Wilson,  or 

order,   four   hundred   thirty-two   and   ^j-   dollars,  value 
received. 

Discounted  at  6^%,  Jan.  6.  JOHN  ALLEN. 

3.  $670.35.  ST.  Louis.  Jan.  6,  1885. 
Ninety  days  from  date  I  promise  to  pay  to  the  order  of 

Peter  Holmes  six  hundred  severity  and  -j^j-  dollars,  value 
received. 

Discounted  at  7%,  Jan.  26.  ROBERT  DAY. 

4.  $1304.90.  CINCINNATI,  Jan.  25,  1885. 
Five  months  after  date  I  promise  to  pay  to  the  order  of 

John  Shannon  thirteen  hundred  four  and  -ffo  dollars,  for 
value  received,  with  interest  at  six  per  cent. 

Discounted  at  4^-%,  March  15.         CHARLES  HILLMAN. 


5.    $2260.  BALTIMORE,  MD.,  June  19,  1885. 

Sixty  days  from  date  I  promise  to  pay  to  the  order  of 
John  Morrison  twenty-two  hundred  and  sixty  dollars,  value 
received. 

Discounted  at  5£%,  July  16.  FRANK  HOWE. 


236  INTEREST  AND  DISCOUNT. 

6.  $645.  AUSTIN,  TEX.,  July  28,  1885. 
Thirty  days  from  date  I  promise  to  pay  to  the  order  of 

John    Moses    six   hundred   and   forty-five    dollars,    value 
received. 

Discounted  at  6%,  Aug.  3.  RICHARD  SMITH. 

7.  $1000.  SAVANNAH,  GA.,  Oct.  4,  1884. 
Six  months  after  date  I  promise  to  pay  to  John  Proctor, 

or  order,  one  thousand  dollars,  value  received,  with  interest 
at  seven  per  cent. 

Discounted  at  8%,  Dec.  31.  JAMES  WHITRIDGE. 

8.  $2912.60.  PHILADELPHIA,  Feb.  19,  1885. 
Ninety  days  after  date  I  promise  to  pay  to  the  order  of 

George  Wright  twenty-nine  hundred  twelve  and  -ffo  dol- 
lars, value  received. 

Discounted  at  6$,  March  1.  PETER  BURKE. 

9.  $455.04.  CHARLESTON,  S.C.,  Sept.  2,  1885. 
Four  months  from  date  I  promise  to  pay  to  the  order  of 

Edmund  Home  four  hundred  fifty-five  and  y^j-  dollars, 
value  received. 

Discounted  at  5|$>,  Sept.  16.  PAUL  WEST. 

10.  $1140.  NEW  ORLEANS,  LA.,  July  1,  1885. 
Ninety  days  after  date  I  promise  to  pay  to  the  order  of 

William   Whitridge    eleven    hundred   and   forty   dollars, 
value  received. 

Discounted  at  7£%,  Aug.  15.  JOHN  CLEMENT. 


11.    $10,089.25.  DENVER,  COL.,  Oct.  14,  1885. 

Ninety  days  after  date  I  promise  to  pay  to  the  order  of 
John  Higgins  ten  thousand  eighty-nine  and  -ffo  dollars, 
value  received. 

Discounted  at  10%,  Dec.  1.  JOHN  KELLEY, 


INTEREST  AND  DISCOUNT.  237 

231.  To  determine  the  face  of  a  note  that  will  yield  a 
given  sum  when  discounted. 

For  how  much  must  a  four-months'  note  without  interest 
be  made  that  it  may  yield  $1000  when  discounted  at  a 
bank  at  6%  ? 

The  discount  on  $1  for  4  mos.  3  dys.  is  $0.0205. 
Proceeds  of  $  1  is  $  1  -  ?  0.0205  =  $  0.9795  =  0.9795  of  $  1. 
Therefore  the  face  required  is  $  1000  -*-  0.9795  =  $  1020.93. 

Ex.  148. 

1 .  Find  the  face  of  a  note  for  30  dys.  that  will  realize 

$600  when  discounted  at  6J%. 

2.  Find  the  face  of  a  note  for  60  dys.  that  will  realize 

$8000  when  discounted  at  8%. 

3.  Find  the  face  of  a  four-months*  note  that  will  realize 

$800  when  discounted  at  5J%. 

4.  Find  the  face  of  a  note  for  90  dys.  that  will  realize 

$1700  when  discounted  at  1%. 

6.    Find  the  face  of  a  two-months'  note  that  will  realize 
$900  when  discounted  at  7T85%. 

6.    Find  the  face  of  a  three-months'  note  that  will  realize 
$2200  when  discounted  at  7%. 

PRESENT  WOBTH  AND  DISCOUNT. 

232,  The  present  worth  of  a  sum  of  money  due  at  the 
end  of  a  fixed  time  is  the  sum  that,  put  at  interest  for  the 
fixed  time,  will  amount  to  the  given  sum. 

Thus,  $100  will  in  2  yrs.,  at  6%,  amount  to  $112.  And  $112  to 
be  paid  at  the  end  of  2  yrs.  is  equal  in  value  to  $  100  paid  now. 
Hence  $100  is  regarded  as  the  present  worth  of  $112  to  be  paid  in 
2  yrs. 


238 


INTEREST  AND   DISCOUNT. 


COMMERCIAL  DISCOUNT. 

233,  Commercial  discount  is  a  reduction  from  the  nominal 
price  or  value  of  anything. 

234.  Price-lists  of  articles  manufactured   and   sold   are 
issued   by  manufacturers   and   wholesale   dealers.     These 
prices  are  subject  to  many  and  various  discounts. 

The  following  bill  will  afford  a  good  illustration  of  this 
discount  : 

JTew  York,  Feb.  1,  1889. 


of  8.   L.  JtfOIflSOJ?  $  CO. 


8  doz.  £olts,  ^3.00 
(Discount,  40,  5,  25, 
11  gro.  Screws,  92.25 
f  and  30, 
6  doz.  Qkest  Handles,  91-50 
40,  5,  25,  17$, 

'-  '4 
13 

00 
74 

$10 
5 
3 

26 
78 
18 

24 
18 

75 

97 

9 
5 

00 
82 

$19 

22 

In  the  first  item,  $3.00  is  the  list  prico  per  dozen  of  the  bolts,  $24 
is  the  gross  price  of  the  8  dozen  bolts,  $10.26  is  the  net  price.  This 
net  price  is  found  by  taking  40%  from  $24,  which  leaves  $14.40; 
then  taking  5%  from  $14.40,  which  leaves  $13.68,  and  then  taking 
25%  from  $13.68,  which  leaves  $  10.26. 

In  the  second  item,  the  gross  price  is  $24.75.  The  f  means  a  dis- 
count of  §  of  the  gross  price,  which  leaves  $  8.25,  and  the  30  means 
there  is  a  discount  of  30%  from  $8.25,  which  leaves  $5.78. 

In  the  third  item,  40%  is  taken  off  the  gross  price,  5%  taken  from 
the  remainder  thus  found,  then  25%  from  the  second  remainder,  and 
then  17  J%  from  the  third  remainder. 

NOTE.  In  finding  17J%,  first  find  10%,  and  then  add  to  it  J  for  5% 
and  i  for  2}%. 


INTEREST  AND   DISCOUNT.  239 

To  combine  two  discounts  so  as  to  form  one  discount,  from  their 
sum  take  one  per  cent  of  their  product.    Thus,  50%  and  10%  =  50  +  10 


Ex.  149. 

Find  the  net  amount  of  a  bill  of  : 

1.  $320  subject  to  a  discount  of  10%  for  cash. 

2.  $1680  subject  to  discounts  of  15  and  10. 

3.  $980  with  15  off. 

4.  $1620  with  20  and  15  off. 

5.  $1440  with  25,  10,  and  5  off. 

6.  $587.50  with  35  and  15  off. 

7.  $1920  with  25  and  72l  off. 

8.  $1530  with  25,  10,  and  5  off. 

9.  $500  with  25,  15,  and  12|  off. 

10.  $870.40  with  30,  22  J,  and  121  off. 

11.  Find  the  net  cash  amount  of  a  bill  of  $1088,  discounts 

being  50  and  10,  and  an  additional  discount  of  5% 
for  cash. 

12.  Find  the  difference  between  a  single  discount  of  55, 

and  successive  discounts  of  40  and  15. 

13.  Find  the  net  cash  amount  of  a  bill  of  $136,  discounts 

being  50,  10,  and  5.     Find  a  single  discount  equiv- 
alent to  these  three  successive  discounts. 

14.  Find  the  net  cash  amount  of  a  bill  of  $164.50,  dis- 

counts being  f  and  30. 

15.  Find  the  net  cash  amount  of  a  bill  of  $15,  discounts 

being  40,  5,  25,  and  17J.     Find  a  single  discount 
equivalent  to  these  four  successive  discounts. 


240  INTEREST  AND  DISCOUNT. 


PABTIAL  PAYMENTS. 

235.  When  settlements  of  accounts  are  made  at  the  expi- 
ration of  a  year  or  less,  it  is  customary  to  reckon  interest 
on  each  item  from  the  time  it  is  due  to  the  time  of  settle- 
ment. 

And  when  partial  payments  are  made  and  indorsed  on 
a  note  that  contains  the  words  with  interest,  provided  the 
note  is  paid  in  full  within  a  year,  it  is  usual  to  compute 
the  interest  on  the  principal,  and  on  each  of  the  payments 
to  the  time  of  settlement. 

Samuel  Paine  buys  of  Edgar  Smith  $400  worth  of  goods 
at  30  dys.  At  the  end  of  3  mos.  he  pays  $200,  and  the 
balance  2  mos.  later.  Find  the  balance. 

The  time  between  the  end  of  30  dys.  and  the  time  of  settlement  is 
4  mos.  Therefore  interest  is  reckoned  on  the  $400  for  4  mos.,  and 
on  the  $  200  for  2  mos. 

$400  +  4  mos.  interest  -  $  408 

$200  +  2  mos.  interest  -    202 

Therefore  balance  due  is  $206 

A  man  holds  a  note  for  $1000,  dated  Jan.  1,  1885,  on 
which  are  indorsed  payments  as  follows:  March  1,  1885, 
$100;  Oct.  1,  1885,  $50;  Nov.  1,  1885,  $800.  What  is 
due  Jan.  1,  1886,  interest  at  6%  ? 

Amount  of  $  1000  for  1  yr.f  at  6%,  is  $1060.00 

Amount  of  $  100  for  10  mos.,  at  6%  =  $  105.00 
Amount  of  $50  for  3  mos.,  at  6%  =  50.75 
Amount  of  $800  for  2  mos.,  at  6%  =  808.00 

963.75 

Balance  due,  $96.25 

This  method  is  in  accordance  with  what  is  called  the 
Merchant's  Kule, 


INTEREST  AND  DISCOUNT.  241 

Ex.  150. 

1.  A  note  for  $3000,  dated  April  1,  1884,  payable  on 

demand,  with  interest  at  7%,  bears  the  following 
indorsements  :  May  6,  $600 ;  July  5,  $676.11 ;  Oct. 
18,  $966.  What  is  due  Jan.  1,  1885  ? 

2.  A  note  for  $1237.50,  dated  April  17,  1884,  payable  on 

demand,  bears  the  following  indorsements :  June  5, 
$253 ;  Aug.  20,  $274.50  ;  Nov.  17,  $420.  What  is 
due  Jan.  1,  1885,  reckoning  interest  at  6%  ? 

3.  A  note  for  $775.50,  dated  May  15,  1884,  payable  on 

demand,  bears  the  following  indorsements:  July  21, 
$150  ;  Oct.  10,  $250;  Feb.  24,  1885,  $100.  What 
is  due  May  15,  1885,  reckoning  interest  at  6%  ? 

4.  A  note  for  $1670.50,  dated  July  1,  1884,  payable  on 

demand,  with  interest  at  6^-%,  bears  the  following 
indorsements:  Aug.  20,  $315;  Sept.  21,  $360.50; 
Oct.  5,  $400;  Dec.  1,  $160.  What  is  due  Jan.  1, 
1885? 

236.  When  a  note  that  contains  the  words  "  with  interest" 
runs  longer  than  a  year,  and  partial  payments  have  been 
made,  the  interest  is  computed  by  a  rule  adopted  by  the 
Supreme  Court  of  the  United  States,  and  therefore  called 

THE  UNITED  STATES  RULE. 

Find  the  amount  of  the  principal  to  the  time  when  the 
payment,  or  sum  oj  the  payments,  equals  or  exceeds  the 
interest. 

From  this  amount  deduct  the  payment  or  sum  of  the  pay- 
ments. 

Consider  the  remainder  as  a  new  principal,  and  proceed 
as  before. 


242  INTEREST  AND   DISCOUNT. 

Ex.  A  note  of  $1520,  dated  May  20,  1884,  and  drawing 
interest  at  6%,  had  payments  indorsed  upon  it  as 
follows:  Oct.  2,  1884,  $300;  Feb.  26,  1885,  $25 ; 
April  2,  1885,  $570;  Aug.  8,  1885,  $600.  Find 
the  amount  due  Dec.  6,  1885. 

yr.       mo*,     djs. 

1884     10      2  $1520  let  principal. 

1884  6    20  .022 

4     12        .022  $33.44  1st  interest. 

1520.00 
1 300.  $1553.44 

300.00  1st  payment. 

1885  2    26  $1253.44  2d  principal 

1884  10      2  .024 

4    24        .024  $25    $30.08  2d  interest. 

$1253.44  2d  principal. 
$25.  .006 

$570    $7.52  3d  interest 

1885  4      2  30.08  2d  interest 
1885      2    26                                 1253.44 

1      6         006  $1291.04 

595.00  2d  &  3d  payments 
f  570  $696.04  3d  principal. 

.021 

1885      8      8  $14.62  4th  interest 

1885      4      2  696.04 

4      6        .021  $710.66 

600.00  4th  payment 
$600.  $110.66  4th  principal. 

.019| 

1885    12      6  $2.18  5th  interest 

1885      8      8  110.66 

3    28       .019$  $112.84    $112.84.  Ana. 

In  the  first  place,  find  the  difference  in  time  between  each  pair  of 
consecutive  dates.  At  the  right  of  the  result  in  each  case  put  the 
corresponding  decimal  multiplier  for  the  interest  at  6%,  and  put  the 
corresponding  payment  below. 


INTEREST  AND   DISCOUNT.  243 

Generally,  it  can  be  determined  mentally  whether  one  or  more 
payments  must  be  taken  to  make  a  sum  equal  to  or  greater  than  the 
interest.  If  two  or  more  payments  are  required,  the  corresponding 
decimal  multipliers  may  be  added,  and  the  result  taken  for  the  mul- 
tiplier. Thus,  it  is  evident  that  .024  of  $1253.44  is  more  than  $25; 
therefore  .024  4-  .006  —  .03  may  be  taken  for  the  multiplier,  which 
will  give  for  the  interest  $37.60.  To  this  the  principal  is  added,  and 
from  the  amount  the  sum  of  the  payments  is  subtracted. 

When  the  rate  is  greater  or  less  than  6%,  the  several  interests 
must  be  increased  or  diminished  according  to  the  given  rate* 

Ex.  151. 

1.  A  note  of  $1000,  dated  Jan.  22,  1884,  and  drawing 

interest  at  6%,  had  payments  indorsed  upon  it  as 
follows  :  May  20, 1884,  $50 ;  July  20, 1884,  $  162.50 ; 
Dec.  23, 1884,  $  72.50.  Find  the  balance  due  March 
1,  1885. 

2.  A  note  of  $3325,  dated  Jan.  15,  1884,  and  drawing 

interest  at  6^-%,  had  payments  indorsed  upon  it  as 
follows :  June  24, 1884,  $100;  Sept.  2, 1884,  $1250; 
Jan.  31,  1885,  $1400.  Find  the  balance  due  May 
12,  1886. 

3.  A  note  of  $2280,  dated  Jan.  22,   1883,  and  drawing 

interest  at  7%,  had  payments  indorsed  upon  it  as 
follows:  Jan.  10, 1884, $1000;  Aug. 31, 1884, $250; 
Jan.  15,  1885,  $600;  March  4,  1885,  $430.  Find 
the  balance  due  June  15,  1885. 


COMPOUND  INTEREST. 

237,    When   a   note   contains  the   words   "with   interest 
annually,"  and  ^the  interest  is  not  paid  at  the  time  it  is  due,  ' 
the  inter-ast  is  ,»miaUy.  added  to  the  principal;    and  new 
principals  are  thus  formed  at  regular  intervals  of  timeT 


244  INTEREST  AND   DISCOUNT. 

238.  The  interest  may  be  compounded  with  the  principal 
(that  is,  made  a  part  of  the  principal)  annually,  semi- annu- 
ally, quarterly,  etc.,  according  to  agreement. 

Ex.  Find  the  compound  interest  of  $800  for  2  yrs.  3  mos. 
15  dys.,  at  7%. 

$800 
.07 

$56  1st  interest 
800 
$856  2d  principal 

.07 

$59.92  2d  interest 
856.00 


$915.92  3d  principal. 
,       0175 

3  mos.  15  dys.       \   6)16.03 
(       2.67 

$18.70  3d  interest 
915.92 


$934.62  amount 

800.00 
$134.62  interest. 

$134.62.  Ans. 

239.  If  the  given  time  be  not  an  integral  number  of  years, 
the  amount  is  found  for  the  number  of  entire  years,  and 
then  the  amount  of  this  for  the  fractional  part  of  a  year. 

Ex.  152. 

I.    Find    the   compound    interest   on   $125   for   3   yrs., 
at 


2.  Find  the  amount  of  $87.50  for  3  yrs.,  at  4%  per  annum, 

at  compound  interest. 

3.  Compare  the  simple  and  compound  interest  on  $21.50, 

•at  the  end  of  4  yrs.,  at  5%. 


INTEREST  AND  DISCOUNT.  245 

4.  What  will  a  debt  of  $4250  amount  to,  if  left  standing 

for   2   yrs.    6  mos.,  at  5%    per  annum,  compound 
interest  ? 

5.  Find  the  compound  interest  on  $104  for  1  yr.  9  mos., 

at  5%. 

6.  Find  the  compound  interest  on  $1800  for  2  yrs.  3  mos. 

15  dys.,  at  3|%. 

7.  Find  the  compound  interest  on  $4500  for  3  yrs.  6  mos., 

at  4%. 

If  the  interest  be  payable  semi-annually,  quarterly,  etc.,  the 
half,  quarter,  etc.,  of  the  rate  per  cent,  must  be  used,  and  the 
amount  obtained  for  each  half-year,  quarter-year,  etc. 

8.  Find  the  compound  interest  on   $4000  for  2  yrs.  6 

mos.,  at   5%    per  annum,   interest  payable   semi- 
annually. 

9.  Find  the  compound  interest  on  $1001.50  for  1  yr. 

3  mos.,  at  6%,  interest  payable  semi-annually. 

10.  Find  the  compound  interest  on  $4000  for  1  yr.  3  mos., 

at  4%  per  annum,  interest  payable  quarterly. 

Ex.  What  principal  will  produce  in  2  yrs.  $650.14,  com- 
pound interest  at  6%  ? 

Amount  of  $1  for  1  yr.,  at  6%,  is  1.06  X  $1. 

Amount  of  $  1  for  2  yrs.  is  1.06  x  1.06  X  $  1  =  (1.06)*  x  $  1. 

That  is,  the  amount  of  $1  for  2  yrs.,  at  6%,  is  $1.1236. 

Interest  is  $  1.1236-$!  -  $0.1236  -  0.1236  of  $1. 

The  principal  required  is  $650.14  -*•  0.1236  =  $5260. 

$5260.  Ans. 

11.  What  principal  will  amount  to  $275.62  in  2  yrs.,  at 

5%  compound  interest? 

12.  What  principal  will  amount  to  $620.32  in  3  yrs.,  at 

6%  compound  interest  ? 


246  INTEREST   AND   DISCOUNT. 

ANNUAL  INTEREST. 

240,  Annual  Interest  is  simple  interest  on  the  principal 
and  on  each  year's  interest  from  the  time  each  interest  is  due 
until  settlement. 

(1)  Find  the  interest  due  Aug.  4,  1885,  on  a  note  dated 
June  4,  1881,  for  $1700,  with  interest  payable 
annually,  at  6%. 

yrs.      moi.  dy§. 

1885    8    4  $1700.00 

1881     6    4  0.25 

4     2  $425.00    Interest  for  4  yrs.  2  mos. 


Annual  interest. 


4.08 
36.72 

$40.80    Interest  on  annual  int. 
425.00 


3    2 
2    2 
1     2 
2 

$1700.00 
.06 

$102.00 
.06 

6    8  =  6f  yrs. 

$6.12 

$465.80    Total  interest  due. 

$465.80.  Ans. 

The  first  year's  interest,  $  102,  remains  overdue  3  yrs.  2 
mos.,  the  second  year's  2  yrs.  2  mos.,  the  third  year's  1  yr. 
2  mos.,  and  the  fourth  year's  2  mos.  Now  the  interest  on 
$  102  for  the  sum  of  these  periods,  6f  yrs.,  is  $  40.80.  Hence 
the  total  interest  is  $465.80. 

13.  Find  the  amount  due  May  17,  1885,  on  a  note  dated 

May  17,  1881,  for  $700,  at  6%  annual  interest. 

14.  Find  the  amount  due  May  27,  1885,  on  a  note  dated 

Jan.  4,  1883,  for  $431,  at  5$%  annual  interest. 

15.  Find  the  amount  due  May  19,  1885,  on  a  note  dated 

Dec.  26,  1881,  for  $612.30,  at  5%  annual  interest. 


INTEREST  AND   DISCOUNT.  247 

16.  Find  the  amount  due  Jan.  16,  1885,  on  a  note  dated 

Jan.  8,  1883,  for  $623.04,  at  5%  annual  interest. 

17.  Find  the  amount  due  Jan.  18,  1885,  on  a  note  dated 

Jan.  8,  1881,  for  $575,  at  6%  annual  interest. 

STOCKS  AND  BONDS. 

241,  The  name  stock  is  applied  to  the  capital  of  banks, 
railroads,  and  other  incorporated  companies. 

The  capital  of  a  company  is  usually  divided  into  shares, 
of  which  the  original  value  is  $100,  or  some  other  fixed 
sum ;  but  the  market-value  at  any  time  is  the  price  per 
share  at  that  time. 

When  the  market-value  of  stock  is  equal  to  its  origi- 
nal value,  it  is  said  to  be  at  par.  In  quotations  of 
stocks,  par  is  generally  represented  by  100 ;  and  when  stock 
is  quoted  at  above  100,  it  is  said  to  be  at  a  premium ;  below 
100,  at  a  discount.  The  premium  or  discount  is  the  dif- 
ference between  the  quotation  and  100. 

Thus,  when  the  price  of  a  stock  on  a  given  day  is  91,  or,  as  it  is 
commonly  expressed,  when  the  stock  is  at  91,  the  meaning  is,  that 
$  100  stock  costs  on  that  day  $91  money  ;  or  that,  if  100  be  the  repre- 
sentative of  any  quantity  of  stock,  91  will  represent  the  correspond- 
ing value  in  money.  In  this  case  the  stock  is  said  to  be  9%  discount. 

The  buying  and  selling  of  stocks  is  conducted  through  the  agency 
of  stock-brokers,  who  receive  a  commission  on  the  stock.  The  com- 
mission is  generally  reckoned  at  \  of  1%  on  the  par  value  of  the  stock. 
Thus,  if  a  broker  buy  stock  for  a  person  at  91,  that  person  pays  91 J. 

(1)  How  much  would  be  received  for  52  shares  of  stock, 
$100  each,  at  89£? 

$  J  per  share  will  represent  the  commission. 
52x$89£    =$4654. 
Commission  =          6.50 

Proceeds      =$4647.50  $4647.50.   Ans. 


248  INTEREST  AND  DISCOUNT. 

(2)  What  amount  of  stock,   at  84f,  including  brokerage, 

may  be  bought  for  $9393.37£? 

Since  $0.84f,  or  0.84£  of  $1,  buys  $1  stock,  the  amount  bought 
for  $9393.37J  will  be  $9393-37*  =  $11,100. 

$11,100.   Ans. 

(3)  What  is  the  quoted  price  of  stock  when  $42,464.25  is 

paid  for  $46,600  stock? 

$46,600  stock  costs  $42,464.25. 

$1  stock  costs  ^J^  of  $42,464.25  =  $0.91J.        gl«     ^ 

Ex.   153. 

1.  Find  the  cost  of  $5000  stock,  at  98. 

2.  Find  the  cost  of  $7800  stock,  at  78$. 

3.  Find  the  cost  of  $20,000  stock,  at  109f 

4.  Find  the  cost  of  $5000  United  States  4%  bonds,  at 

121. 

5.  Mr.  Jones  owns  20  United  States  4%  bonds  of  $1000 

each.  The  interest  on  these  bonds  is  paid  quarterly. 
How  much  interest  does  Mr.  Jones  receive  every 
quarter  ? 

6.  Find  the  cost  of  20  shares  of  Boston  and  Maine  Kail- 

road  stock,  at  174. 

7.  How  much  of  United  States  4%  bonds  may  be  bought 

for  $6305,  at  1211? 

8.  How  much  of  Northern  Pacific  6%  bonds,  selling  at 

102|,  may  be  bought  for  $10,275? 

9.  How  many  shares  ($100  each)  of  Old  Colony  Railroad, 

at  137£,  may  be  bought  for  $1650? 


INTEREST   AND   DISCOUNT.  249 

10.  How  many  shares  of  railroad  stock,  at  91^,  may  be 

bought  for  $8474.62|? 

11.  What  must  be  the  price  of  stock  in  order  that  $9200 

stock  may  be  bought  for  $8970? 

12.  What  must  be  the  price  of  stock  in  order  that  $11,600 

stock  may  be  bought  for  $8729? 

13.  If  $3000  stock  is  bought  for  $2748.75,  what  is  the 

price  of  the  stock  ? 

14.  What  income  will  be  derived  from  $15,000  of  5% 

bonds  ? 

15.  Find  the  income  from  $9000  of  6%  stock. 

16.  How  much  will  a  person  receive  from  $18,800  railroad 

stock,  if  a  dividend  of  1%  be  declared? 

17.  What  income  will  be  derived  from  $30,000  of  4% 

bonds? 

Ex.  How  much   4%    stock   must   be  bought  to  give  an 
income  of  $320? 

Since  $0.04  is  derived  from  $1  stock,  $320  will  be  derived 
from  as  many  times  $  1  as  $0.04  is  contained  in  $320.  $320  ^ 
$0.04  =  $8000. 

$8000.    Am. 

18.  How   much   4%    stock  must  be  bought  to  give  an 

income  of  $2400? 

19.  A  person  receives  $343  as  his  quarterly  dividend  from 

a  7%  stock.     How  much  stock  does  he  hold? 

20.  Find  the  entire  income  of  a  person  whose  property 

consists  of  $6000  of  6%  stock  and  $16,400  of  7% 
stock. 

21.  Find  the  rate  of  dividend  paid  by  a  railroad  when  a 

holder  of  246  shares  receives  $1722. 


250  INTEREST  AND  DISCOUNT. 

22.  Find  the  rate  per  cent  at  which  $22,200  will  yield  a 

semi-annual  return  of  $990. 

Ex.  If  $5125  is  invested  in  6%  stock,  at  102J,  what  income 
will  be  obtained  ? 

$1  stock  costs  1.025  of  $1. 

Hence  $5125   will   be  the   cost  of  $5125-*-  1.025  =  $5000 
stock.     And  6%  of  $  5000  =  $  300. 

$300.  Ans. 

23.  Find  the  income  on  $39,000  invested  in  4%  stock,  at 

91. 

24.  Find  the  income  on  $7000  invested  in  4%  stock,  at 


25.  Find  the  income  on  $13,600  invested  in  7%  stock,  at 

130. 

26.  A  person  invests  $14,280  in  railroad  stock,  at  127^. 

What  will   he   receive   if  a   dividend   of  3J%  be 
declared  ? 

27.  Find  the  income  on  $14,000  when  invested  in  8% 

stock,  at  103J. 

Ex.  If  a  person  buys  5%  stock  at  120,  what  rate  of  interest 
does  he  receive  on  his  money  invested  ? 

$100  stock  costs  $120.     $100  stock  pays  $5.     Hence  the 
$120  invested  yields  $5. 

Therefore,  the  rate  of  interest  is  —  =  0.04  J,  or  4J% 


.  Ans. 

28.  If  an  8%  stock  is  worth  150,  what  rate  of  interest  will 

a  purchaser  receive  on  his  money  ? 

29.  If  a  10%  stock  is  worth  175,  what  rate  of  interest  will 

a  purchaser  receive  on  his  money  ? 


INTEKEST  AttD  DISCOUNT.  251 

30.  If  a  9%  stock  is  worth  170,  what  rate  of  interest  will 

a  purchaser  receive  on  his  money  ? 

31.  If  a  4%  stock  is  worth  70,  what  rate  of  interest  will  a 

purchaser  receive  on  his  money  ? 

32.  If  a  3%  stock  is  worth  65,  what  rate  of  interest  will  a 

purchaser  receive  on  his  money  ? 

Ex.  Find  the  sum  required  for  an  investment  in  a  4%  stock, 
at  98£,  to  produce  an  income  of  $200  a  year. 
$4  are  received  from  $100  stock. 

Hence  $200  will  be  received  from  —  X  $100  stock  =  $5000 
stock. 

$100  stock  costs  $  98 J. 

Therefore  $5000  stock  will  cost  50  x  $  98  J  =  $  4925. 

$4925.   Ana. 

33.  How  much  money  must  be  invested  in  8%   stock,  at 

92,  to  produce  $400  income? 

34.  How  much  money  must  be  invested  in  a  3%  stock,  at 

87^-,  to  produce  an  income  of  $250? 

35.  A  person  bought  some  bank  stock  at  107,  and  received 

$265  when  a  5%  dividend  was  declared  by  the  bank. 
How  much  money  had  he  invested  ? 

36.  A  person  buys  some  6%   railroad  stock  at  75,  and 

receives  $750  income.     How  much  money  has  he 
invested  ?  9 

Ex.  What  must  be  the  price  of  a  5%  stock  in  order  that  a 
buyer  may  receive  6%  on  his  investment  ? 

$100  must  be  invested  to  produce  $6. 

Hence  \  of  $100  =  $83J  must  be  invested  to  produce  $5. 

Therefore  the  price  of  the  5%  stock  must  be  83 J. 

Ans. 


252  INTEREST  AND  DISCOUNT. 

37.  What  must  be  the  price  of  a  6%  stock  in  order  that  a 

buyer  may  receive  7%  on  his  investment? 

38.  What  must  be  the  price  of  an  8%  stock  in  order  that 

a  buyer  may  receive  6%  on  his  investment? 

39.  A  person  invested  $5710  in  bank  stock  when  the  stock 

was  at  142£.      What  per  cent  dividend  is  declared, 
if  he  receives  $300? 

40.  A  person  receives  5%  interest  on  his  money  by  invest- 

ing in  some  six  per  cent,  stock.     At  what  price  did 
he  buy  it? 

41.  What  must  be  the  price  of  a  7%  stock  in  order  that  a 

buyer  may  receive  6%  on  his  investment? 

EXCHANGE. 

242.  A  bank  draft  or  bill  of  exchange  is  a  written  order 
directing  one  person  to  pay  a  specified  sum  of  money  to 
another. 

243.  A  commercial  draft  is  a  draft  payable  at  a  specified 
time  after  sight  (or  date). 

When  the  person  on  whom  a  commercial  draft  is  drawn  accepts 
the  draft,  ho  writes  the  word  "  Accepted,"  with  the  date,  across  the 
fact,  and  signs  his  name.  The  draft  is  then  called  an  acceptance, 
and  the  acceptor  is  responsible  for  its  payment. 

An  acceptance  is  of  the  nature  of  a  promissory  note,  the  acceptor 
and  maker  having  respectively  the  same  responsibility  for  payment 
as  the  maker  and  indorser  of  a  promissory  note. 

244.  The  system  of  paying  money  to  persons  at  a  dis- 
tance by  transmitting  bank    drafts  or  bills  of  exchange 
instead  of  money  is  called  exchange. 

When  a  bank  draft  can  be  bought  for  its  face,  it  is  said  to  be  at  par. 
When  the  cost  is  less  than  the  face,  it  is  said  to  be  at  a.  discount;  and 
when  the  cost  is  more  than  the  face,  it  is  said  to  be  at  a  premium. 


INTEREST  AND  DISCOUNT.  253 

Ex.   154. 

Ex.  Find  the  cost  of  a  draft  on  New  York  for  $1000,  at  \ 
of  1%  premium. 

£%  of  $1000  =  $2.50  (premium). 
$1000  +  $2.50  =  $1002.50  (cost). 

$1002.50.  Ans. 

1.  Find  the  cost  of  a  draft  on  New  York  for  $1200,  at  \ 

of  1%  discount. 

2.  Find  the  cost  of  a  draft  on  St.  Louis  for  $2000,  at  J  of 

1%  premium. 

3.  Find  the  cost  of  a  draft  on  New  Orleans  for  $2400,  at 

•J-%  premium. 

4.  Find  the  cost  of  a  draft  on  Chicago  for  $3200,  at  f  % 

discount. 

Ex.  Find  the  cost  of  a  draft  on  Cincinnati  for  $1000,  pay- 
able in  30  dys.  after  sight,  exchange  being  •£% 
premium,  and  interest  6%. 

$1000.00 
0.0055  of  $1000  =        $5.50*di8count  for  33  dys. 

$994.50  cost  of  draft  at  par. 
0.005    of  $1000-       ^  5.00  premium. 

$999.50  cost  of  draft. 

5.  Find  the  cost  of  a  draft  for  $800,  payable  30  dys.  after 

sight,  when  exchange  \^\°/o  premium,  and  interest 
6%. 

6.  Find  the  cost  of  a  draft  for  $1900,  payable  in  30  dys., 

when  exchange  is  at  par,  and  interest  4|-%. 

7.  Find  the  cost  of  a  draft  for  $1450,  payable  in  GO  dys., 

when  exchange  i-s  ^%  discount,  and  interest  5%. 

8.  Find  the  cost  of  a  draft  for  $1000,  payable  60  dys. 

after  sight,   when    exchange   is   -|%    discount,  and 
interest  7%. 


CHAPTER  XII. 

PROPORTION. 

245,  The  relative  magnitude  of  two  numbers  is  called 
their  ratio  when  expressed  by  the  fraction  which  has  the 
first  number  for  numerator,  and  the  second  number  for 
denominator. 

Thus  the  ratio  of  2  to  3,  commonly  written  2  :  3,  is  expressed  by 
the  fraction  J. 

246,  The  first  term  of  a  ratio  is  called  the  antecedent, 
and  the  second  term  the  consequent. 

247,  If  both  terms  of  a  ratio  be  multiplied  or  divided  by 
the  same  number,  .the  ratio  is  not  altered. 

Thus,  if  both  terms  of  the  ratio  2J :  3J  be  multiplied  by  6,  the 
resulting  ratio  is  15 :  20,  and  fhe  two  ratios  are  equal,  for  -*  «=»  JJ. 

^3 

Since  J$  =»  f ,  the  simplest  expression  for  2| :  3J  is  3  :  4. 

248,  If  the  numerator  and  denominator  are  interchanged, 
the  fraction  is   said  to  be  inverted;  likewise,  if  the  ante- 
cedent and  consequent  of  a  ratio  are  interchanged,    the 
resulting  ratio  is  called  the  inverse  of  the  given  ratio. 

Thus,  if  the  fraction  £  is  inverted,  the  resulting  fraction  is  J,  and 
the  inverse  of  the  ratio  4  :  5  is  5  :  4. 

249,  If  two  quantities  are  expressed  in  the  same  unit, 
their  ratio  will  be  the  same  as  the  ratio  of  the  two  numbers 
by  which  they  are  expressed. 

Thus  the  quantity  $  7  is  the  same  fraction  of  $  9  as  7  is  of  9. 


PROPORTION.  255 


250.  Since  ratio  is  simply  relative  magnitude,  two  quan- 
tities different  in  kind  cannot  form  the  terms  of  a  ratio  ; 
and  two  quantities  of  the  same  kind  must  be  expressed  in  a 
common  unit  before  they  can  form  the  terms  of  a  ratio. 

Thus  no  ratio  exists  between  $5  and  20  dys. ;  and  the  ratio  of  3  t. 
to  5000  Ibs.  can  be  expressed  only  when  both  quantities  are  written 
as  tons  or  pounds. 

251.  When  two  ratios  are  equal,  the  four  terms  are  said 
to  be  in  proportion,  and  are  called  proportionals. 

Thus  6,  3,  18,  9  are  proportionals ;  for  f  =  -1/. 

252.  A  proportion  is  written  by  putting  the  sign  =  or  a 
double  colon  between  the  ratios. 

Thus  6 :  3  =  18  :  9,  or  6 :  3 : :  18 :  9,  means,  and  is  read,  the  ratio  of 
6  to  3  is  equal  to  the  ratio  of  18  to  9. 

253.  The  j£rs£  and  last  terms  of  a  proportion  are  called 
the  extremes,  and  the  two  middle  terms  are  called  the  means. 

254.  Test  of  a  proportion.     When  four  numbers  are  pro- 
portionals, the  product  of  the  extremes  is  equal  to  the 
product  of  the  means. 

This  is  seen  to  be  true  by  expressing  the  ratios  in  the 
form  of  fractions,  and  multiplying  both  by  the  product  of 
the  denominators. 

Thus  the  proportion  5  :  3  : :  15  :  9  may  be  written  $  =  -^ ;  and,  if 
both  be  multiplied  by  3  X  9,  the  result  will  be  5  X  9  =  3  X  15. 

255.  Either  extreme,  therefore,  will  be  equal  to  the  prod- 
uct of  the  means  divided  by  the  other  extreme ;  and  either 
mean  will  be  equal  to  the  product  of  the  extremes  divided 
by  the  other  mean.     Hence,  if  three  terms  of  a  proportion 
be  given,  the  fourth  may  be  found. 


256  PROPORTION. 


(1)  What  number  is  to  4  as  3  is  to  6? 

This  may  be  written    What  number  =  ?  ? 
Multiply  both  sides  of  the  equation  by  4. 

The  result  is,  What  number 

6 

2.  Ans. 

(2)  20  is  to  24  as  what  number  is  to  30? 

This  may  be  written    gO  a  What  number  ? 
24  30 

Multiply  by  30,  20x30  -  What  number  ? 

25.  Ans. 

(3)  18  is  to  32  as  45  is  to  what  number? 

This  may  be  written    —  = — ? 

32      What  number 

As  these  fractions  are  equal,  their  reciprocals  are  equal ; 

that  is,  —  =  What  number  ^ 

18  45 


Multiply  by  45,  32x45 

18 


80.  Ans. 


256,  When  three  terms  of  a  proportion  are  given,  the 
method  of  finding  the  fourth  term  is  called  the  Eule  of  Three, 

It  is  usual  to  arrange  the  quantities  (that  is,  to  state  the 
question)  so  that  the  quantity  required  for  the  answer  may 
be  the  fourth  term.  Hence  the  quantity  which  corresponds 
to  that  of  the  required  answer  must  be  the  third  term. 

(1)    If  5  t.  of  hay  cost  $87.50,  what  will  21  t.  cost? 

Since  the  cost  of  21  t.  is  required,  $  87.50  is  the  third  term. 
Since  21  t.  will  cost  more  than  5  t.,  21  t.  is  the  second  term 
and  5  t.  the  first  term. 

That  is,  5  t  :  21  t. :  :  $87.50  :  What  quantity? 


PROPORTION.  257 


A  difficulty  presents  itself  here,  inasmuch  as  no  meaning  can 
be  given  to  the  product  of  the  means  ($87.50  multiplied  by  21 1.). 
Since,  however,  the  ratio  of  5  t. :  21  t.  =  the  ratio  of  5 :  21,  the 
ratio  5  :  21  may  be  substituted  for  5  t. :  21  t. 

Then  5 :  21 : :  $87.50 :  What  quantity  ? 

That  is,  What  quantity  =  gl.X$  87.50  ? 

$367.50.  Ans. 

(2)  When  a  post  11.5  ft.  high  casts  a  shadow  on  level 
ground  17.4  ft.  long,  a  neighboring  steeple  casts  a 
shadow  63.7  yds.  long.  How  high  is  the  steeple? 

Height  is  required ;  the  height  11.5  ft.  is  therefore  the  third 
term. 

Since  the  shadow  of  the  steeple  is  the  longer,  the  height  of  the 
steeple  must  be  the  greater ;  therefore  the  second  term  must  be 
the  greater  of  the  two  remaining  quantities  expressed  in  the 
same  unit.  63.7  yds.  =  191.1  ft. 

Shadow.        Shadow.  Height.        Height. 

17.4  ft. :  191.1  ft.  : :  11.5  ft. :  What? 
or,  17.4      : 191.1       ::  11.5  ft. :  What? 

That  is.  height  of  steeple  =  19L1  *  1L5  ft  -  126.3  ft. 

126.3  ft.  Ans. 

257,    In  solving  problems  by  the  Bule  of  Three, 

Make  that  quantity  which  is  of  the  same  kind  as  the 
required  answer  the  third  term. 

The  numbers  by  which  the  other  two  quantities  are  ex- 
pressed, when  expressed  in  a  common  unit,  will  be  the  first 
and  second  terms. 

If,  from  the  nature  of  the  question,  the  answer  will  be 
greater  than  the  third  term,  make  the  greater  of  these  two 
numbers  the  second  term  ;  if  less,  make  the  smaller  of  these 
numbers  the  second  term,  and  the  other  the  first  term. 

Divide  the  product  of  the  second  and  third  terms  by  the 
first  term,  and  the  quotient  will  be  the  answer  required. 


258  PROPORTION, 


Ex.  155. 

1.  An  express-train  runs  40  mi.  in  64  mm.     At  the  same 

rate,  how  many  miles  will  it  run  in  24  min.  ? 

2.  If  110  A.  produce  200  hhds.  of  sugar,  how  many  hogs- 

heads will  176  A.  produce? 

3.  If  48  reapers  cut  20  A.  in  a  given  time,  how  many 

acres  will  156  reapers  cut  in  the  same  time  ? 

4.  If  20  reapers  can  cut  a  field  in  6  dys.,  in  how  many 

days  will  30  reapers  do  it? 

5.  The  number  of  copies  in  the  first  edition  of  the  "  Lady 

of  the  Lake  "  was  2050,  and  was  to  the  number  in 
the  second  as  41  to  69.  Find  the  number  in  the 
second  edition. 

6.  The  length  of  the  steamer-track   from   Liverpool  to 

Quebec  is  2502  mi.,  and  is  to  that  from  Liverpool 
to  Boston  as  139  is  to  155.  Find  the  length  of  the 
track  from  Liverpool  to  Boston. 

7.  If  a  steamer  from  Liverpool  to  Portland  makes  the 

passage  of  2750  mi.  in  5-f-  dys.,  in  how  many  days, 
at  the  same  rate,  would  the  passage  of  2980  mi.  from 
Liverpool  to  New  York  have  been  made  ? 

8.  If  a  person  can  walk  8^-  mi.  in  2J-  hrs.,  how  many  miles 

can  he  walk  in  3  J  hrs.  ? 

9.  If  the  shadow  of  a  staff  3  ft.  7  in.  high  is  4  ft.  9  in., 

find  the  height  of  a  steeple  whose  shadow,  is  158  ft. 
4  in. 

10  A  train,  at  the  rate  of  25f  mi.  an  hour,  goes  a  certain 
distance  in  3£  hrs.  In  how  many  hours  will  one  at 
the  rate  of  24J  mi.  an  hour  go  the  same  distance  ? 


PROPORTION.  259 


11.  The  ratio  of  the  diameter  to  the  circumference  of  a 

circle  was  given  by  Metius  as  113  :  355.     Find  the 
circumference  of  a  fly-wheel  10  ft.  in  diameter. 

12.  Find  the  horse-power  of  an    engine   that   can   raise 

11,200  Ibs.  of  coal  in  an  hour  from  a  pit  whose  depth 
is  396  ft. 

NOTE.    The  labor  necessary  to  raise  1  Ib.  through  1  ft.  is  called  the 
unit  of  work ;  and  a  horse  can  do  33,000  units  of  work  a  minute. 

Therefore  one  horse-power  =  33,000  units  of  work,  and  396  X  112QQ 

33000  X  60 
=»  the  horse-power  required. 

13.  If  1000  sq.  yds.  of  a  field  produce  a  load  of  hay,  how 

many  such  loads  will  25  A.  of  the  field  produce  ? 

14.  If  a  train  runs  177  mi.  120  rds.  in  3  hrs.  56^-  min., 

what  is  the  rate  per  hour? 

16.    If  136  masons  can  build  a  fort  in  28  dys.,  how  many 
men  will  be  required  to  build  it  in  8  dys.  ? 

16.  There  are  provisions  in  a  fort  sufficient  to  support 

4000  soldiers  for  3  mos.     How  many  must  be  sent 
away  to  make  them  last  8  mos.  ? 

17.  A  coach  travels  7-J-  mi.  an  hour.    How  many  miles  will 

it  go  between  a  quarter  past  ten  A.M.  and  a  quarter 
to  six  P.M.  ? 

18.  The  expense  of  making  the  hay  on  5  A.  135  sq.  rds.  is 

$29.08.     What  is  the  expense  per  acre? 

19.  If  300  laborers  can  make  an  embankment  in  48  dys., 

how  many  more  days  would  be  required  if  the  num- 
ber of  men  is  diminished  by  60  ? 

20.  If  2.45  tons  of  straw  cost  $22.75,  how  many  tons  can 

be  bought  for  $11.70? 


260  PROPORTION. 


COMPOUND  PROPORTION. 

258,  A  ratio  is  said  to  be  compounded  of  two  or  more 
given  ratios,  when  it  is  expressed  by  a  fraction  which  is  the 
product  of  the  fractions  representing  the  given  ratios. 

Thus  the  ratios  2:3  and  7  :  11  are  represented  by  the  fractions 
f  and  -fa  ;  and  the  ratio  14  :  33,  which  is  represented  by  Jf  (the  prod- 
uct of  J  and  -fa\  is  said  to  be  compounded  of  the  ratios  2  :  3  and  7:11. 


259,  A  proportion  which  has  one  of  its  ratios  a  compound 
ratio  is  called  a  compound  proportion, 

In  stating  problems  in  compound  proportion  the  quantity 
which  corresponds  to  the  answer  required  is  made  the  third 
term.  Each  pair  of  the  remaining  quantities  is  then  con- 
sidered separately  with  reference  to  the  answer  required. 
The  process  will  be  understood  by  the  following  example  : 

If  4  men  mow  15  A.  in  5  dys.  of  14  hrs.,  in  how  many 
days  of  13  hrs.  can  7  men  mow  19^-  A.  ? 

As  the  answer  is  to  be  in  days,  make  5  dys.  the  third  term. 

I.  Will  it  require  more  or  less  days  for  7  men  to  mow  15  A.  than 
it  did  for  4  men  ?     Evidently  less. 

Therefore  make  7  the  first  term  and  4  the  second. 

II.  Will  it  require  more  or  less  days  for  the  same  number  of  men 
to  mow  19}  A.  than  it  did  to  mow  15  A.  ?     Evidently  more. 

Therefore  make  15  the  first  term  and  19}  the  second. 

III.  Will  it  require  more  or  less  days  of  13  hrs.  to  mow  the  same 
number  of  acres  than  it  did  of  14  hrs.?     Evidently  more. 

Therefore  make  13  the  first  term  and  14  the  second. 
Hence  the  statement  is 

7:4 

15:  19.5:  :  5  days:  what? 
13:14 

4  x  19.5  x  14  x  5  days 
7  X  15  x  13 

This,  simplified  by  cancellation,  gives  4  days. 


PROPORTION.  261 


Ex.  156. 

1.  If  13  bu.  of  oats  serve  3  horses  for  11  dys.,  how  many 

bushels  will  serve  7  horses  for  12  dys.  ? 

2.  If  a  traveller  walks  140  mi.  in  8  dys.,  walking  7  hrs.  a 

day,  how  many  miles  can  he  walk  in  12  dys.  of  8 
hrs.  each  ? 

3.  If  4  masons  build  27  yds.  of  wall  in  5  dys.,  working  9 

hrs.  a  day,  in  how  many  days  will  32  masons  build 
81  yds.  of  a  similar  wall,  if  they  work  10  hrs.  a  day? 

4.  A  bootmaker  who  employs  15  men  fills  an  order  for 

25  doz.  pairs  of  boots  in  4  wks.  In  how  many  days 
can  he  make  45  pairs  if  he  employs  18  men  ? 

5.  If  a  family,  by  using  2  gas-burners  7£  hrs.  a  day,  pays 

$6  a  quarter  when  gas  is  $2.40  per  1000  cu.  ft., 
what  will  a  family  using  3  burners  4  hrs.  a  day  pay 
per  quarter  when  gas  is  $1.80  per  1000  cu.  ft.  ? 

6.  If  330  slices  T\  of  an  inch  thick  are  obtained  from  12 

rounds  of  beef,  how  many  similar  rounds  will  be 
required  for  495  slices  £  of  an  inch  thick  ? 

7.  If  5  horses  eat  8  bu.  14  qts.  of  oats  in  9  dys.,  how 

many  days,  at  the  same  rate,  will  66  bu.  30  qts.  last 
17  horses? 

8.  If  a  man  walks  600  mi.  in  25  dys.,  walking  8  hrs.  a 

day,  in  how  many  days  will  he  walk  330  mi.,  walk- 
ing 10  hrs.  a  day  ? 

9.  If  a  pane  of  glass  18  in.  long  and  12^  in.  wide  costs  20 

cts.,  what  will  be  the  cost,  at  the  same  rate,  of  a 
pane  22-1-  in.  ]ong  an(j  15  |n  wide? 

10.  If  18  men  can  dig  a  trench  200  yds.  long,  3  yds.  wide, 
and  2  yds.  deep,  in  6  dys.  of  10  hrs.  each,  in  how 
many  days  of  8  hrs.  each  will  10  men  dig  a  trench 
100  yds.  long,  4  yds.  wide,  and  3  yds.  deep? 


262  PROPORTION. 


PROPORTIONAL  PARTS. 

260.  If  it  be  required  to  divide  a  quantity  into  parts 
proportional  to  3,  4,  5,  the  numbers  3,  4,  5  may  be  taken 
as  representatives  of  the  parts,  and  then  the  whole  quantity 
will  be  represented  by  3  +  4  +  5  ;  that  is,  by  12. 

(1)  Divide  $391  into  parts  proportional  to  5,  7,  and  11. 

The  whole  quantity  will  be  represented  by  5  +  7  +  11  =  23. 
Therefore  the  respective  parts  will  be  ^,  ^,  £J  of  $391. 

$85,  $119,  $187.  Ans. 

(2)  Divide  $248  into  parts  proportional  to  fa,  -fa,  fa. 

Multiply  the  fractions  by  150,  the  L.C.M.  of  their  denomina- 
tors. The  results  are  15,  10,  6.  Hence  the  parts  will  be  repre- 
sented by  the  numbers  15,  10,  6,  and  the  whole  by  31. 

Therefore  the  respective  parts  will  be  jf,  J£,  -ft  of  $248. 

$120,  $80,  $48.   Ans. 

Ex.  157. 

1.  Divide  1200  into  parts  proportional  to  11,  12,  13,  14. 

2.  Divide  390  into  parts  proportional  to  -£-,  £,  £. 

3.  Divide  a  profit  of  $689  among  3  partners,  of  whom  the 

first  owns  -fa,  the  second  -fa,  and  the  third  ^  of  the 
joint  stock. 

4.  Four  men  invest  $450,  $230,  $190,  $110  respectively 

in  a  joint  business.  Find  their  respective  liabilities 
in  a  loss  of  $313.60. 

5.  Three  partners  claim  respectively  -J-,  -J-|-,  and  -fa  of 

$1260.     Give  to  each  his  proportional  share. 

6.  An  analysis  of  dissolved  bones   gives   the   following 

results  for  every  100  parts.  Water,  13.97  ;  organic 
matter,  15.71;  soluble  phosphates,  21.63;  insoluble 
phosphates,  11.43;  sulphate  of  lime,  15.83;  sulphuric 
acid,  15.63;  alkaline  salts,  1.10;  silica,  etc.,  the 
remainder.  Find  the  number  of  pounds  of  each  in 
a  ton  of  dissolved  bones. 


PROPORTION.  263 


PAETNERSHIP. 

261,  Partnership  is  separated  into  simple  and  compound. 
In  simple  partnership  the  capital  of  each  partner  is  invested 
for  the  same  time.  In  compound  partnership  the  time  for 
which  the  capital  of  each  partner  is  invested  is  taken  into 
account,  as  well  as  the  amount  of  the  capital ;  and  the  divi- 
sion of  profits  and  losses  is  made  proportionally  to  the 
amount  of  the  capital  and  the  time  it  is  invested. 

A  and  B  enter  into  partnership.  A  puts  in  $2000  for 
2  yrs.,  and  B  puts  in  $3000  for  1  yr.  Their  profits  are 
$  1400.  What  is  the  share  of  each  ? 

The  use  of  $2000  for  2  yrs.  is  equivalent  to  2  X  $2000  for  1  yr. 
Hence  their  profits  must  be  divided  in  the  ratio  $4000  to  $3000; 
that  is,  4  :  3. 

Ex.  158. 

1.  Three  drovers  rent  a  field  of  9  A.,  at  $5  an  acre.     A 

puts  in  6  cows  for  2  mos ;  B,  9  cows  for  1  mo.  ;  and 
0,  12  cows  for  2  mos.  How  much  should  each  pay  ? 

2.  In  a  co-partnership  A  contributed  $400  for  9  mos. ;  B, 

$350  for  8  mos. ;  and  C,  $600  for  2  mos.  Divide  a 
gain  of  $570  among  them. 

3.  At  the  end  of  12  mos.  A,  B,  and  C,  having  a  joint 

capital  of  $6000,  find  they  have  lost  $625.  As 
capital  of  $2500  has  been  in  the  business  for  12  mos., 
B's  of  $1500  for  8  mos.,  and  C's  of  $2000  for  4  mos. 
Divide  the  loss  among  them. 

4.  A  and  B  enter  into  partnership,  A  with  $1800,  and 

B  with  $900.  At  the  end  of  8  mos.  B  adds  $300  to 
his  capital.  Divide  a  profit  of  $840  between  them, 
at  the  end  of  the  year. 


264  PROPORTION. 


AVERAGES. 

262,    If  a  dozen  eggs  weigh  1  Ib.  8  oz.,  what  is  their 
average  weight? 

Since  the  12  eggs  weigh  1  Ib.  8  oz.,  that  is,  24  oz.,  the  average 
weight  of  an  egg  will  be  ^j  of  24  oz.  «=  2  oz. 

Ex.  159. 

1.  A  merchant  mixes  3  Ibs.  of  coffee   worth  27  cts.  a 

pound,  2  Ibs.  worth  35  cts.,  and  1  Ib.  worth  41  cts. 
What  is  the  mixture  worth  a  pound  ? 

2.  What  is  the  cost  of  a  gallon  of  a  mixture  containing 

7  gals,  worth  $1.35  a  gallon,  5  gals,  worth  $1.05  a 
gallon,  and  water  enough  to  make  the  whole  mix- 
ture 15  gals.  ? 

3.  Of  32  candidates  for  office,  3  were  20  yrs.  old,  4  were 

21,  12  were  22,  12  were  23,  and  1,  24.  What  was 
the  average  age  of  the  candidates  ? 

4.  A   bankrupt  owes   A   $962.50,   B,   $3487,    and    C, 

$12,686.50.  His  estate,  after  paying  expenses  of 
settlement,  is  $3427.20.  How  much  can  he  pay  on 
a  dollar  ? 

5.  A  grocer  buys  106  Ibs.  of  tea,  at  80  cts.  per  pound, 

75  Ibs.,  at  $1.24  per  pound,  and  94  Ibs.,  at  $1.30 
per  pound,  and  mixes  the  three  lots  together.  At 
what  price  per  pound  must  he  sell  the  mixture  so 
as  to  make  10%  on  his  outlay? 

6.  In  what  proportions  must  oils  worth  $1.25  a  gallon 

and  80  cts.  a  gallon  be  mixed  to  make  a  mixture 

worth  $1.00  a  gallon? 

HINT.  The  loss  on  the  $1.25  oil  is  25  cts.  a  gallon.  The  gain 
on  the  80  ct.  oil  is  20  cts.  a  gallon.  Therefore  there  must  be 
more  of  the  80  ct.  oil  taken  than  of  the  $1.25  oil,  and  in  the 
ratio  of  25  :  20  or  5  :  4. 


PROPORTION.  265 


7.  In  what  proportion  must  oils  worth  $1.20  and  60  cts. 

a  gallon  be  mixed,  so  that  the  mixture  may  be  worth 
70  cts.  a  gallon  ? 

8.  Solder  is  composed  of  tin  and  lead.    If  a  solder  weighs 

10.44  times  as  much  as  an  equal  bulk  of  water, 
while  tin  weighs  7.29,  and  lead  11.35  as  much,  find 
the  weight  of  each  metal  in  a  pound  of  solder. 

AVERAGE  OF  PAYMENTS. 

A  has  given  to  B  notes  as  follows :  $  250,  due  in  3  mos. ; 
$400,  due  in  6  mos. ;  $700,  due  in  8  mos.  He  wishes  to 
pay  them  all  at  one  time.  In  how  many  months  shall  the 
entire  payment  be  made  ? 

The  use  of   $250  for  3  mos.  equals  the  use  of   $750  for  1  mo. 

The  use  of   $400  for  6  mos.  equals  the  use  of  $2400  for  1  mo. 

The  use  of   $700  for  8  mos.  equals  the  use  of  $5600  for  1  mo. 
$1350  $8750  for  1  mo. 

The  question  is,  for  how  many  months  is  the  use  of  $1350  equal 
to  the  use  of  $8750  for  1  mo.  ? 

The  answer  required  is  {Jf  $  mos.  =•  6£J  mos. 

ty%  mos.  Ans. 

9.  Find  the  equated  time  for  the  payment  of  $300  due 

in  3  mos.,  $500  due  in  6  mos.,  $200  due  in  9  mos. 

10.  A  owes  B  $50  payable  in  6  mos.,  $60  payable  in  8 

mos.,  and  $90  payable  in  4  mos.  Find  the  equated 
time  of  payment. 

11.  A  owes  B  $1000,  payable  at  the  end  of  9  mos.     He 

pays  $200  at  the  end  of  3  mos.  and  $300  at  the  end 
of  8  mos.  When  is  the  balance  due  ? 

12.  On  the  first  day  of  January,  A  purchases  of  B  $200 

worth  of  goods  on  3  mos.  credit,  and  $500  worth  on 
4  mos.  credit,  and  gives  one  note  in  payment.  When 
does  the  note  become  due? 


CHAPTER  XIII. 

POWERS  AND  ROOTS. 

263.  The  square  of  a  number  is  the  product  of  two  fac- 
tors, each  equal  to  this  number. 

Thus  the  squares  of  1,  2,  3,    4,    5,    6,    7,    8,    9,    10, 
are  1,  4,  9,  16,  25,  36,  49,  64,  81,  100 

264.  The  square  root  of  a  number  is  one  of  the  two  equal 
factors  of  the  number. 

Thus  the  square  roots  of  1,  4,  9,  16,  25,  36,  49,  64,  81,  100, 
are  1,2,3,    4,   5,    6,    7,    8,    9,    10. 

265.  The  square  root  of  a  number  is  indicated  by  the 
radical  sign  -^/,  or  by  the  fraction  -J-  written  above  and  to 
the  right  of  the  number. 

266.  Since  35  =  30+5,  the  square  of  35  may  be  obtained 
as  follows : 

30+5 

30+5  30'=   900 

30'+    (30x5)  2(30x5)=   300 

(30x5)  +  5*  5'=     25 

30'+ 2  (30x5) +  5*  =1225 

267.  Hence,  since  every  number  consisting  of  two  or  more 
figures  may  be  regarded  as  composed  of  tens  and  units, 

The  square  of  a  number  will  contain  the  square  of  the 
tens  +  twice  the  tens  X  the  units  +  the  square  of  the  units. 


POWERS    AND    ROOTS.  267 

SQUARE  ROOT. 

The  first  step  in  extracting  the  square  root  of  a 
number  is  to  mark  off  the  figures  of  the  number  in  groups. 

Since  1  =  I2,  100  =  102,  10,000  =  1002,  and  so  on,  it  is  evident  that 
the  square  root  of  any  number  between  1  and  100  lies  between  1  and 
10 ;  of  any  number  between  100  and  10,000  lies  between  10  and  100. 
In  other  words,  the  square  root  of  any  number  expressed  by  one  or 
two  figures  is  a  number  of  one  figure ;  of  any  number  expressed  by 
three  or  four  figures  is  a  number  of  two  figures,  and  so  on. 

If,  therefore,  an  integral  number  be  divided  into  groups  of  two  fig- 
ures each,  from  the  right  to  the  left,  the  number  of  figures  in  the  root 
will  be  equal  to  the  number  of  groups  of  figures.  The  last  group  to 
the  left  may  consist  of  only  one  figure. 

Find  the  square  root  of  1225. 

The  first  group  12,  contains  the  square  of  the  tens' 
12  25  (35     number  of  the  root. 

9  The  greatest  square  in  12  is  9,  and  the  square  root 

65)  3  25  of  9  is  3.     Hence  3  is  the  tens*  figure  of  the  root. 

3  25  The  square  of  the  tens  is  subtracted,  and  the 

remainder,  contains  twice  the  tens  X  the  units  +  the 
square  of  the  units.  Twice  the  3  tens  is  6  tens,  and  6  tens  is  con- 
tained in  the  32  tens  of  the  remainder  5  times.  Hence  5  is  the  units' 
figure  of  the  root.  Since  twice  the  tens  X  the  units  +  the  square  of 
the  units  is  equal  to  (twice  the  tens  +  the  units)  X  the  units,  the  5 
units  are  annexed  to  the  6  tens,  and  the  result,  65,  is  multiplied  by  5. 

269,  The  same  method  will  apply  to  numbers  of  more  than 
two  groups  of  figures,  by  considering  the  part  of  the  root  al- 
ready found  as  so  many  tens  with  respect  to  the  next  figure. 

Extract  the  square  root  of  7890481. 

7  89  04  81  (2809          When  the  third  group,  04,  is  brought 

4  down,  and  the  divisor,  56,  formed,  the  next 
48)  3  89  figure  of  the  root  is  0,  because  56  is  not  con- 

3  34  tained  in  50.     Therefore,  0  is  placed  both 

5  04  81  *n  ^e  r00^  an(^  ^e  Divisor,  and  the  next 

5  04  81  two  figures»  81,  are  brought  down. 


268  POWERS  AND   ROOTS. 

270.  If  the  square  root  of  a  number  have  decimal  places, 
the  number  itself  will  have  twice  as  many. 

Thus,  if  0.11  be  the  square  root  of  some  number,  the  number  will 
be  (O.ll)2  -  0.11  X  0.11  =  0.0121.  Hence,  if  a  given  number  contain 
a  decimal,  we  divide  it  into  groups  of  two  figures  each,  by  beginning 
at  the  decimal  point  and  marking  toward  the  left  for  the  integral 
number,  and  toward  the  right  for  the  decimal.  We  must  be  careful 
to  have  the  last  group  on  the  right  of  the  decimal  point  contain  two 
figures,  annexing  a  cipher  when  necessary. 

Extract  the  square  root  of  52.2729. 

'52.2729(7.23 
49 

149)3  97  Jt  wil1  te  8een  frcm  tte  ?roups  of  figures 

9  £4  that  the  root  will  have  ene  integral  and  two 

1443)1329  ^imal  places. 

4329 

271.  If  a  number  is  not  a  perfect  square,  ciphers  may  be 
annexed,  and  an  approximate  value  of  the  root  found. 

Extract  to  six  places  of  decimals  the  square  root  of  19. 
19000000(4.358899 

16 

83)3  00 

o  40  In   this   example,  after   finding  four 

r figures  of  the  root,  the  other  three  are 

865)  51  00  foun(i  by   common  division.      The   rule 

^  ^  in  such  cases  is,  that  one  less  than  the 

8708/  7  75  00  number  of  figures  already  obtained  may 

6  96  64  be  found  without  error  by  division,  the 

8716)  78  360  divisor  to  be  employed  being  twice  the 

69  728  part  of  the  root  already  found. 
86320 
78444 
78760 

272.  The  square  root  of  a  common  fraction  is  found  by 
extracting  the  square  roots  of  the  numerator  and  denomi- 


POWERS   AND   ROOTS. 


269 


nator.  But,  when  the  denominator  is  not  a  perfect  square, 
it  is  best  to  reduce  the  fraction  to  a  decimal  and  then 
extract  the  root. 

Ex.  160. 
Find  the  square  roots  of : 


1.  4225. 

5.  15.7609. 

9.  0.025. 

13.  T6^-. 

2.  31.36. 

6.  0.180625. 

10.  28.75. 

14-  «f 

3.  50625. 

7.  0.001296. 

11.  0.009. 

15.  f. 

4.  401956. 

8.  0.042849. 

12.  0.081. 

16.  4. 

The  side  of  a  square  is  found  by  extracting  the  square 
root  of  its  area. 

17.  A  rectangle  is  972yds.  long  and  432  yds.  wide.     Find 

the  side  of  a  square  which  has  the  same  area  as  the 
rectangle. 

18.  Find  in  yards  the  length  of 

the  side  of  a  square  field 

containing  27  A.  12  sq.  rds. 

1  sq.  yd. 

In  a  right  triangle,  the  square 
on  the  hypotenuse  {AC)  is  equal 
to  the  sum  of  the  squares  on  the 
two  legs. 

Hence  hypotenuse  «=  square  root  of 
sum  of  squares  on  the  legs  ;  and  one  leg  =  square  root  of  difference 
of  squares  on  the  other  two  sides. 

19.  Base  =  39,  perpendicular  —  52  ;  find  hypotenuse. 

20.  Base  =  35,  hypotenuse  —  91 ;  find  perpendicular. 

21.  Perpendicular  =  72,  hypotenuse  =  75  ;  find  base. 

22.  A  cord  287  ft.  long  is  stretched  from  the  top  of  a  flag- 

pole 63  ft.  high ;    find  the  distance  of  the  end  in 
contact  with  the  ground  from  the  base  of  the  pole. 


270  POWERS   AND   ROOTS. 

TJie  length  of  the  diagonal  of  a  room  is  the  square  root  of 
the  sum  of  the  squares  of  the  length,  breadth,  and  height. 

23.  Find  the  diagonal  of  a  room  28  ft.  long,  21  ft.  wide, 

and  12  ft.  high. 

24.  Find  the  diagonal  of  a  hall  50  ft.  long,  30  ft.  wide,  and 

15  ft.  high. 

CUBE  ROOT. 

273,  The  cube  of  a  number  is  the  product  of  three  factors, 
each  equal  to  the  number. 

The  cubes  of  1,  2,    3,    4,      5,      G,      7,      8,      9,      10, 
are  1,  8,  27,  64,  125,  216,  343,  512,  729,  1000. 

274,  The  cube  root  of  a  number  is  one  of  the  three  equal 
factors  of  the  number. 

Thus  the  cube  roots  of  1,  8,  27,  64,  125,  216,  343,  512,  729,  1000, 
are  1,2,    3,    4,      5,      6,      7,      8,      9,      10. 

275,  The  cube  root  of  a  number  is  indicated  by  •{/,  or  by 
the   fraction   -^     written  above  and   to  the  right  of  the 
number. 

Thus,  v/343,  or  343$,  means  the  cube  root  of  343, 

276,  Since  35  =  30  +  5,  the  cube  of  35  may  be  obtained 
thus: 

30+5 
30+5 

30'+    (30x5)  30s-  27,000 

+    (30x5)  +  52  3(302x5)=13,500 

302+2(30x5)  +  52  3(30  X  52)  =    2,250 

30+5  _  5s-  _  125 

303+2(302x5)+    (30  x52)  42,875 

(302x5)  +  2(30x52) 


303+  3(302  x  5)  +  3(30  x  52)  +  5s 


POWERS  AND  ROOTS.  271 

Hence  the  cube  of  any  number  composed  of  tens  and 
units  contains  four  parts : 

I.    The  cube  of  the  tens. 

II.    Three  times  the  product  of  the  square  of  the  tens  by 
the  units. 

III.  Three  times  the  product  of  the  tens  by  tht  square  of 
the  units. 

IV.  The  cube  of  the  units. 

277,  In  extracting  the  cube  root  of  a  number,  the  first 
step  is  to  mark  off  the  figures  of  the  number  in  groups. 

Since  1  =  1s,  1000  =  103, 1,000,000  =  1003,  and  so  on,  it  follows  that 
the  cube  root  of  any  number  between  1  and  1000,  that  is,  of  any  num- 
ber that  has  one,  two,  or  three  figures,  is  a  number  of  one  figure ;  and 
that  the  cube  root  of  any  number  between  1000  and  1,000,000,  that 
is,  of  any  number  that  has  four,  five,  or  six  figures,  is  a  number  of  two 
figures,  and  so  on. 

.  If,  therefore,  an  integral  number  be  divided  into  groups  of  three 
figures  each,  from  right  to  left,  the  number  of  figures  in  the  root  will 
be  equal  to  the  number  of  groups.  The  last  group  to  the  left  may 
consist  of  one,  two,  or  three  figures. 

Extract  the  cube  root  of  42875. 

42  875  (35       Since  42875  consists  of  two 
27  groups,  the   cube   root   will 


15  875          consist  of  two  figures. 

The  first  group,  42,  contains 

the  cube  of  the  tens'  number 

of  the  root. 
15  875  The  greatest   cube  in  42 


3  X  30*  =  2700 

3  X  (30x5)=   450 

5'=     25 

3175  , 

is  27,  and  the  cube  root  of 
27  is  3.     Hence  3  is  the  tens'  figure  of  the  root. 

The  remainder,  15875,  resulting  from  subtracting  the  cube  of  the 
tens,  will  contain  three  times  the  product  of  the  square  of  the  tens  by 
the  units  +  three  times  the  product  of  the  tens  by  the  square  of  the 
units  +  the  cube  of  the  units. 

Each  of  these  three  parts  contains  the  units'  number  as  a  factor. 


272  POWERS   AND   ROOTS. 

Hence  the  15875  consists  of  two  factors,  one  of  which  is  the  units' 
number  of  the  root ;  and  the  other  factor  is  three  times  the  square  of 
the  tens  +  three  times  the  product  of  the  tens  by  the  square  of  the 
units  -f  the  square  of  the  units.  The  larger  part  of  this  second  factor 
is  three  times  the  square  of  the  tens. 

And,  if  the  158  hundreds  of  the  remainder  be  divided  by  the 
3  x  302  =-  27  hundreds,  the  quotient  will  be  the  units'  number  of  the 
root. 

The  second  factor  can  now  be  completed  by  adding  to  the  2700 
3  x  (30  X  5)  =  450  and  5*  =  25. 

278.  The  same  method  will  apply  to  numbers  of  more 
than  two  groups  of  figures,  by  considering  the  part  of  the 
root  already  found  as  so  many  tens  with  respect  to  the 
next  figure  of  the  root. 

Extract  the  cube  root  of  57512456. 

57512456(386 

3  x  30'  =     2700 

3  X  (30  x  8)  =       720 

8'  =         64 


3484 


2  640  456 

3  x  3801  =  433200  ' 
3  X  (380  x  6)  =     6840 
6'=         36 


440076 


27872 


2  640  456 


279,  If  the  cube  root  of  a  number  have  decimal  places, 
the  number  itself  will  have  three  times  as  many. 

Thus,  if  0.11  be  the  cube  root  of  a  number,  the  number  is  0.11  X 
0.11  x  0.11  =  0.001331.  Hence,  if  a  given  number  contain  a  decimal, 
we  divide  the  figures  of  the  number  into  groups  of  three  figures  each, 
by  beginning  at  the  decimal-point  and  marking  toward  the  left  for 
the  integral  number,  and  toward  the  right  for  the  decimal.  We  must 
be  careful  to  have  the  last  group  on  the  right  of  the  decimal-point 
contain  three  figures,  annexing  ciphers  when  necessary. 


POWERS  AND  ROOTS. 


273 


Extract  the  cube  root  of  187.149248. 

187.149248(5.72 
125 
8x50'  =7500 


3  X  (50  X  7)  =  1050 

7*=  49 

8599 


62149 


60193 


3  x570'  =  974700 

3  X  (570  X  2)  =     3420 

2a=  4 


978124 


1  956  248 


1  956  248 


It  will  be  seen  from  the  groups  of  figures  that  the  root  will  have 
one  integral  and  two  decimal  places,  and  therefore  the  decimal-point 
must  be  placed  in  the  root  as  soon  as  one  figure  of  the  root  is  obtained. 

280.  If  the  given  number  be  not  a  perfect  cube,  ciphers 
may  be  annexed,  and  a  value  of  the  root  may  be  found  as 
near  to  the  true  value  as  we  please. 

Extract  the  cube  root  of  1250.6894. 

1250.689400(10.77 
1 

3x10*=     300  |    250 

Since  300  is  not  contained  in  200,  the  next  figure  of  the  root  will 
beO. 

250  689 


3  x  1002  =  30000 
3  X  (100x7)=  2100 

72  = 49 

32149 


225  043 


3  x  10702  =  3434700 
3  X  (1070  x  7)  =  22470 

72  = 49 

3457219 


25  646  400 


24  200  533 


1  445  867 


274 


POWERS   AND   ROOTS. 


281.    The  following  method  very  much  shortens  the  work 
in  long  examples. 

Extract  the  cube  root  of  5  to  five  places  of  decimals. 


5.000(1.70997 
1 


3  X  102  =  300 
3  (10x7)  =  210 
7'  =  J9-j 
559  [ 
259  J 


4000 


3913 


3  x  17002  =  8670000 
3(1700x9)-   45900 
9'  =  _  81} 
8715981  I 
45981  J 


3  x  1709'  =  8762043 


87  000  000 


78  443  829 


85561710 
7  885  8387 


670  33230 
613  34301 


After  the  first  two  figures  of  the  root  are  found,  the  next  trial  divi- 
sor is  obtained  by  bringing  down  the  sum  of  the  210  and  49  obtained 
in  completing  the  preceding  divisor,  then  adding  the  three  lines  con- 
nected by  the  brace,  and  annexing  two  ciphers  to  the  result. 

It  is  seen  at  a  glance  that,  when  the  trial  divisor  is  increased  by 
3  times  the  17  tens  of  the  root,  it  will  be  greater  than  87000 ;  so  that 
0  is  placed  in  the  root,  and  3  X  17002  is  obtained  by  annexing  two 
ciphers  to  the  86700.  Again :  the  trial  divisor  is  obtained  by  bringing 
down  the  sum  of  the  45900  and  81,  which  was  obtained  in  completing 
the  preceding  divisor,  then  adding  the  three  lines  connected  by  the 
brace,  and  annexing  two  ciphers  to  the  result. 

The  last  two  figures  of  the  root  are  found  by  division.  The  rule 
in  such  cases  is,  that  two  less  than  the  number  of  figures  already 
obtained  may  be  found  without  error  by  division,  the  divisor  to  be 
employed  being  three  times  the  square  of  the  part  of  the  root  already 
found. 


POWERS   AND   ROOTS. 


275 


282.  The  cube  root  of  a  common  fraction  is  found  by 
taking  the  cube  roots  of  the  numerator  and  denominator ; 
but,  if  the  denominator  be  not  a  perfect  cube,  it  is  best  to 
reduce  the  fraction  to  a  decimal,  and  then  extract  the  root. 


Ex.  161. 


Find  the  cube  roots  of: 


9.  12396.8834. 

10.  0.00027. 

11.  0.00008. 

12.  277.2738. 

Find  the 


1  29791.  5.  53157376. 

2  357911.  6.  62099136. 

3.  148877.  7.  41.421736. 

4.  103823.  8.  12.812904. 

17.  The  liter  contains  61.027  cu.  in. 

cube  containing  a  liter. 

18.  The  edges  of  a  rectangular  solid  are  154 

70  ft.  7  in.,  53  ft    1  in.     Find  the  edge 
equivalent  to  it. 

The  square  of  (30  +  5)  =  30*  -f  2  (30  X  5)  +  52. 

The  302  may  be  represented  by  a  square  (Fig.  1)  30  in. 

The  2(30  X  5)  may  be  represented  by  two  strips  30  in. 
in  wide,  of  Fig.  2,  which  are  added  to  two  adjacent  sides 

The  52  may  be  represented  by  the  small  square  of  Fig. 
to  make  Fig.  2  a  complete  square. 


13. 


15.  U- 

16.  f. 

side  of  a 

ft.  11  in., 
of  a  cube 


?266. 
on  a  side, 
long  and  5 
of  Fig.  1. 
3  required 


Fig.  1. 


Fig.  2. 


Fig.  8. 


In  extracting  the  square  root  of  1225,  the  large  square,  which  is  30 
in.  on  a  side,  is  first  removed,  and  a  surface  of  325  sq.  in.  remains. 

This  surface  consists  of  two  equal  rectangles,  each  30  in.  long,  and 
a  small  square  whose  side  is  equal  to  the  width  of  the  rectangles. 

The  width  of  the  rectangles  is  found  by  dividing  the  325  sq.  in.  by 
the  sum  of  their  lengths,  that  is,  by  60,  which  gives  5  ia. 


276 


POWERS   AND   ROOTS. 


Hence  the  entire  length  of  the  surfaces  added  is  30  in.  +  30  in. 
+  5  in.  =  G5  in.,  and  the  width  is  5  in. 

Therefore  the  total  area  is  (65  X  5)  =  325  sq.  in. 


The  cube  of  (30  +  5)  =  303  +  3  (302  X  5)  +  3  (30  X  52)  +  5s.       g  392. 

The  303  may  be  represented  by  a  cube  whose  edge  is  30  in.  (Fig.  1). 

The  3  (302  X  5)  may  be  represented  by  three  rectangular  solids, 
each  30  in.  long,  30  in.  wide,  and  5  in.  thick,  to  be  added  to  three 
adjacent  faces  of  Fig.  1. 

The  3(30  X  52)  may  be  represented  by  three  equal  rectangular 
solids,  30  in.  long,  5  in.  wide,  and  5  in.  thick,  to  be  added  to  Fig.  2. 

The  53  may  be  represented  by  the  small  cube  required  to  complete 
the  cube  of  Fig.  3. 


Fig. 


Fig.  S. 


Fig.  4. 


In  extracting  the  cube  root  of  42875,  the  large  cube  (Fig.  1),  whose 
edge  is  30  in.,  is  first  removed. 

There  remain  (42875  -  27000)  cu.  in.  =  15875  cu.  in. 

The  greater  part  of  this  is  contained  in  the  three  rectangular  sol- 
ids which  are  added  to  Fig.  1,  and  which  are  each  30  in.  long  and 
30  in.  wide. 

The  thickness  of  these  solids  is  found  by  dividing  the  15875  cu.  in. 
by  the  sum  of  the  three  faces,  each  of  which  is  30  in.  square ;  that  is, 
by  2700  sq.  in.  The  result  is  5  in. 

There  are  also  the  three  rectangular  solids  which  are  added  to 
Fig.  2,  and  which  are  30  in.  long  and  5  in.  wide ;  and  a  cube  which 
is  added  to  Fig.  3,  and  which  is  5  in.  long  and  5  in.  wide. 

Hence  the  sum  of  the  products  of  two  dimensions  of  all  these 
solids  is 

For  the  larger  rectangular  solids,  3(30  X  30)  sq.  in.  =  2700  sq.  in. 

For  the  smaller  rectangular  solids,  3  (30  X  5)  sq.  in.  =    450  sq.  in. 

For  the  small  cube,  (5  x  5)  sq.  in.  = 25  sq.  in. 

3175  sq.  in. 

This  number  multiplied  by  the  third  dimension  gives  (5  x  3175) 
cu.  in.  —  15,875  cu.  in. 


POWERS   AND    ROOTS.  277 

283,    In  bodies  of  the  same  shape, 

Two  corresponding  lines  are  in  the  same  ratio  as  any  other 
two. 

The  ratio  of  two  corresponding  surfaces  is  the  square  of 
the  ratio  of  two  corresponding  lines. 

The  ratio  of  two  corresponding  volumes  is  the  cube  of  the 
ratio  of  two  corresponding  lines. 

Conversely, 

The  ratio  of  two  corresponding  lines  is  the  square  root  of 
the  ratio  of  two  corresponding  surfaces,  and  the  cube  root 
of  the  ratio  of  two  corresponding  volumes. 

Ex.  162. 

1.  The  volume  of  a  rectangular  solid  is  1728  cu.  in.     The 

volume  of  a  similar  solid  is  3375  cu.  in.     Find  the 
ratio  of  two  corresponding  edges. 

2.  The  surface  of  a  solid  is  600  sq.  in.     What  is  the  sur- 

face of  a  similar  solid  whose  edges  are  twice  as  great  ? 

3.  If  the  volumes  of  two  similar  solids  be  100  cu.  in.  and 

1000  cu.   in.   respectively,  find   the  ratio  of  their 
heights  to  the  nearest  thousandth  of  an  inch. 

4.  If  two  hills  have  the  same  shape,  and  one  is  2700  ft. 

high,  while  the  other  is  3600  ft.  high,  find  the  ratio 
of  their  surfaces,  and  also  the  ratio  of  their  volumes. 

5.  A  bushel  measure  and  a  peck  measure  are  of  the  same 

shape.     Find  the  ratio  of  their  heights. 

6.  The  surfaces  of  two  hills  having  the  same  shape  are 

as  25  :  16.     Find  the  ratio  of  their  heights. 

7.  Of  two  similar  solids,  the  volume  of  the  larger  is  1££ 

of  that  of  the   smaller.     Find   the    ratio   of  their 
heights  ;  find  also  the  ratio  of  their  bases. 

8.  The  equatorial  diameter  of  the  earth  is  7926  mi.    Find 

that  of  Venus  whose  volume  is  0.953  of  the  volume 
of  the  earth. 


CHAPTER  XIV. 

MENSURATION. 

(PRACTICAL  RULES.) 

284.  A  surface  has  two  dimensions :  length  and  breadth, 

285.  A  solid  has  three  dimensions :  length,  breadth,  and 
thickness. 

286.  The  area  of  a  surface  is  the  number  of  units  of  sur- 
face which  it  contains,  the  unit  of  surface  being  a  square 
which  has  a  linear  unit  for  each  of  its  dimensions. 

287.  The  volume  of  a  solid  is  the  number  of  units  of 
volume   which   it   contains,  the   unit   of  volume   being  a 
cube  which  has  a  linear  unit  for  each  of  its  three  dimen- 
sions. 

288.  In  writing  the  dimensions  of  surfaces  and  solids,  the 
sign  X  is  used  for  the  word  by,  an  accent  (')  for  the  word 
feet,  and  two  accents  (")  for  the  word  inches.     Thus,  the 
dimensions  of  the  floor  of  a  room,  15  feet  6  inches  long, 
13  feet  8  inches  wide,  are  denoted  by  15'  6"  X  13f  8".     The 
dimensions   of  a   brick,  8  inches  long,  4  inches  wide,  2-J 
inches  thick,  are  denoted  by  8"  X  4"  X  2£". 

289.  Rectangle.      The   area   of  a   rectangle   equals   the 
product  of  its  length  arid  breadth.     (See  page  165.) 

290.  The  perimeter  of  a  rectangle  or  of  any  other  sur- 
face figure  is  the  sum  of  the  lengths  of  the  lines  which 
bound  it. 


MENSURATION.  279 


Ex.  163. 

1.  The  floor  of  a  room  is  a  rectangle  15f  6"  X  18'.     Find 

its  perimeter  and  its  area. 

2.  The  ceiling  of  a  room  is  a  rectangle  16f  X  20'.     Find 

its  perimeter  and  its  area. 

3.  A  rectangular  field  is  60  rods  X  80  rods.      Find  its 

area  in  acres,  and  the  cost  of  fencing  it  at  $1.50  a  rod. 

4.  How  many  boards   12  ft.   long  will  be  required   to 

inclose  a  square  field  48  rds.  on  a  side  with  a  fence 
4  boards  high  ?  How  many  acres  are  there  in  the 
field? 

291.  Carpeting.  Carpeting  is  sold  by  the  yard  in  length. 
The  common  widths  are  a  yard,  and  three-quarters  of  a 
yard.  It  will  be  remembered  that  in  determining  the  num- 
ber of  yards  of  carpet  for  a  room,  we  first  decide  whether 
the  strips  shall  run  lengthwise  or  across  the  room,  and  then 
find  the  number  of  strips  needed.  The  number  of  yards  in 
a  strip,  including  the  allowance  for  waste  in  matching  the 
pattern,  multiplied  by  the  number  of  strips  will  give  the 
required  number  of  yards.  (See  page  169.) 

5.  How  many  yards  of  carpeting  I  of  a  yard  wide  will  be 

required  for  a  floor  20f  X  17'  6",  if  the  strips  run 
lengthwise,  and  if  there  is  a  waste  of  9  in.  a  strip 
in  matching  the  pattern  ? 

6.  How  many  yards   of  carpeting  1  yd.  wide  will  be 

required  for  a  room  18'  4"  X  17'  8",  if  the  strips  run 
lengthwise  of  the  room,  and  if  there  is  a  waste  of 
8  in.  a  strip  in  matching  the  pattern?  Find  the 
cost  of  carpeting  the  room  if  the  carpet  is  worth 
85  cents  a  yard,  and  10  cents  a  yard  is  paid  for 
making  and  laying. 


280  MENSURATION. 


7.  Find  the  cost  of  the  carpet  for  a  room  19'  8"  X  17'  10", 
if  the  carpet  is  J  of  a  yard  wide  and  costs  $1.75  a 
yard,  the  strips  running  across  the  room,  and  9  in.  a 
strip  being  wasted  in  matching  the  pattern. 

292.  Plastering.  The  unit  for  measuring  painting,  plas- 
tering, and  paving  is  the  square  yard.  The  practice  in 
painting  and  plastering  is  to  find  the  total  area  within  the 
bounding  lines  of  the  work,  to  deduct  from  this  amount 
half  the  area  of  all  doors,  windows,  and  other  openings, 
and  to  take  as  the  net  area  the  nearest  whole  number  of 
square  yards  in  the  remainder. 

Ex.  A  rectangular  room  is  15' X  13' 4"  X  9f.  The  base- 
board is  1  foot  high  ;  there  is  a  door  7'  4"  X  4',  and 
two  windows  6'  X  4f  each.  Find  the  cost  of  plaster- 
ing the  walls  and  ceiling  at  18  cents  a  square  yard. 

Perimeter  of  room  =  2  X  15'  +  2  X  13'  4"  =  56'  4". 
Height  of  room  above  baseboard      =  9'  —  1'  =  8'. 
Total  area  of  walls  =    8  X  56J          =  450$  sq.  ft. 
Area  of  ceiling        =  15  X  13  J         =  200    sq.  ft. 

Area  of  walls  and  ceiling 
Height  of  door  above  baseboard 
Area  of  door  above  baseboard 
Area  of  2  windows  =  2  X  6'  X  4' 

Area  of  door  and  windows 

Half  the  area  of  door  and  windows  =  i 

Area  allowed  is 


[  sq.  ft.  -  36§  sq.  ft.  =  614  sq.  ft.  =  AJA  =  68  sq.  yds. 

68  X  18  cents  =  $  12.24.  Am. 

Find  the  cost  of  plastering  the  walls  and  ceiling  of  a 
room  17' 4"  X  15f  8"  X  10' 4",  at  20  cents  per  square 
yard,  if  10  sq  yds.  are  deducted  for  doors,  window?, 
and  baseboard. 


MENSURATION  281 


9.  Find  the  cost  of  whitening  the  walls  and  ceiling  of  a 
room  16'  6"  X  15'  6"  X  9'  6",  at  five  cents  per  square 
yard,  deducting  12  sq.  yds.  for  doors,  windows,  and 
baseboard. 

10.  Find  the  cost  of  plastering  a  room  18'  X  15'  X  10f,  at 

30  cents  per  square  yard,  if  the  room  contains  one 
door  7'  6"  X  4',  three  windows  each  6'  X  4',  and  a 
baseboard  one  foot  high  around  the  room. 

293.  Wall  Paper.  Wall  paper  is  18  in.  wide  and  is  sold 
in  single  rolls  8  yds.  long,  or  in  double  rolls  16  yds.  long. 
In  estimating  the  number  of  rolls  of  paper  required  for  a 
room  of  ordinary  height,  find  the  number  of  feet  in  the 
perimeter  of  the  room,  leaving  out  the  widths  of  the  doors 
and  windows,  and  allow  a  double  roll  or  two  single  rolls 
for  every  7  ft. 

Ex.  How  many  double  rolls  of  paper  will  be  required  for 
a  room  of  ordinary  height,  18'  X  16',  with  one  door, 
and  four  windows,  each  4  ft.  wide  ? 

Perimeter  of  room  =  2  x  18'  +  2  x  16'  =  68' 
Width  of  door  and  windows  =  5  x  4'  =  20' 

Deducting  door  and  windows  =  48' 

^-  =  7.  7  double  rolls.  Ans. 

11.  How  many  double  rolls  of  paper  will  be  required  for 

a  room  of  ordinary  height,  18' 4"  X  16' 6",  with  two 
doors  and  three  windows,  each  4  ft.  wide? 

12.  Find  the  cost  of  paper  at  25  cents  a  single  roll,  and 

bordering  at  8  cents  a  yard,  for  a  room  of  ordinary 
height,  17' 9"  X  17' 3",  allowing  for  one  door  and 
four  windows,  each  4' 2".  (No  allowance  for  doors 
and  windows  is  made  for  the  bordering.) 


282  MENSURATION. 


13.  Find  the  cost  of  paper  at  50  cents  a  single  roll  for  a 

room  of  ordinary  height,  20' 8"  X  17' 6",  with  two 
doors  and  three  windows,  each  4'  2"  wide. 

294.  Laths.     Laths  are  put  up  in  bundles,  100  pieces, 
each  4  ft.  long,  and  a  bundle  is  estimated  to  cover  5  sq. 
yds.     In  estimating  the  number  of  bundles  of  laths,  deduct 
the  whole  area  of  all  openings. 

Ex.  How  many  bundles  of  laths  will  be  required  for  the 
ceiling  of  a  room  36  ft.  square  ? 

36  ft.  =  12  yds. 

Hence  ceiling  contains  12x12=  144  sq.  yds. 

^  =  29.  29  bundles.  Ans. 

14.  How  many  bundles  of  laths  will  be  required  for  the 

ceiling  and  walls  of  a  room  26  ft.  square,  14  ft. 
high,  allowing  20  sq.  yds  for  doors,  windows,  and 
baseboard  ? 

15.  How  many  bundles  of  laths  are  required  for  the  ceiling 

and  walls  of  a  room  28'  X  32'  and  16'  high,  allowing 
for  three  windows  8r  X  3'  6"  each,  and  two  doors 
8'  X  4f  2"  each,  and  a  baseboard  1  ft.  high  ? 

295.  Clapboards.     Clapboards  are  4  ft.  long  and  are  laid 
3^-  or  4  in.  to  the  weather. 

Ex.  Find  the  number  of  clapboards  required  for  the  front 
of  a  house  42  ft.  long  and  22  ft.  high,  allowing 
100  sq.  ft.  for  doors  and  windows,  and  adding  10  % 
for  waste. 

3J  in.  =  5i  ft.  =  &  ft. 

1 L 

4  X  -ff  =  f  f  =  1J  sq.  ft.  for  each  clapboard. 

42  x  22  =  924  sq.  ft. 

924  sq.  ft.  -  100  sq.  ft.  =  824  sq.  ft. 


MENSURATION.  283 


—  =  f  of  824  =  706. 

10%  of  706  =  71. 

706  +  71  =  777.  Ans. 

16.  How  many  clapboards  will  be  required  for  the  front 

of  a  house  40  ft.  long  and  20  ft.  high,  allowing 
96  sq.  ft.  for  doors  and  windows,  and  adding  10  % 
for  waste  ? 

296.  Boofing  and  Flooring.  The  unit  of  measure  for 
roofing  and  flooring  is  a  square  containing  100  sq.  ft. 
Shingles  are  16  in.  long,  and  are  estimated  to  average 
4  in.  wide,  so  that  a  shingle  laid  4-J-  in.  to  the  weather 
would  cover  18  sq.  in.,  and  8  shingles  would  be  required 
for  1  sq.  ft.  At  this  rate  800  shingles  would  cover  a 
square,  but  to  allow  for  waste  it  is  usual  to  reckon  1000 
shingles  to  the  square.  It  is  found,  however,  in  practice, 
that  1000  shingles  of  the  best  quality,  laid  4£  in.  to  the 
weather,  will  cover  about  120  sq.  ft. 

17.  Allowing   1000   shingles   for   120  sq.  ft.,  how  many 

thousand  would  be  required  to  cover  the  pitched 
roof  of  a  house  44  ft.  long,  if  the  width  of  each  side 
of  the  roof  is  24  ft.  ? 

18.  Allowing  1000  shingles  for  110  sq.  ft.,  how  many  thou- 

sand would  be  required  to  cover  the  pitched  roof 
of  a  building  54  ft.  long,  if  the  width  of  each  side 
of  the  roof  is  28  ft.  ? 

19.  How  many  slates  at  3  to  the  square  foot  will  be  re- 

quired to  cover  28  squares  of  roof? 

20.  The   floor  of  a  gymnasium  is  100'  X  60'.     Find  the 

cost  of  birch  for  the  floor  at  $40  a  thousand,  adding 
20%  for  waste. 


284 


MENSURATION. 


297.    Triangles. 
0 


D 


A  triangle  is  a  plane  figure  bounded  by 
three  straight  lines.  Thus,  the  figure 
ABC  is  a  triangle.  The  side  AB 
is  called  the  base ;  the  corner  C oppo- 
site the  base,  the  vertex ;  and  the  per- 

.  pendicular  CD,  drawn  from  Cto  AB, 
the  altitude. 


298.   To  find  the  area  of  a  triangle. 
Take  one-half  the  product  of  the  base  by  the  altitude. 
NOTE.    The  area  of  a  right  triangle  equals  one-half  the  product  of 
the  base  and  the  perpendicular. 

If  the  lengths  of  the  three  sides  of  a  triangle  are  given, 
the  area  is  found  as  follows  : 

From  the  half-sum  of  the  sides  subtract  each  side  sepa- 
rately. Find  the  continued  product  of  the  half-sum  and 
the  three  remainders.  The  square  root  of  the  product  is 
equal  to  the  area  of  the  triangle. 

Ex.    Find  the  area  of  a  triangle  having  a  base  16  ft.  and 
altitude  10  ft. 


Area 


-  =  80.          80  sq.  ft.  Am. 


Ex.    Find  the  area  of  a  triangle  if  the  sides  are  6,  8,  and 
12  ft. 


Area  =  Vl3  x  7  X  5  x  1  =  21.3         21.3  sq.  ft.  Ans. 
Find  the  area  of  a  triangle,  having  given : 

21.  Base  30' 6",  altitude  12'  6". 

22.  Base  148  rds.,  altitude  60  rds. 

23.  Base  10  chains  40  links,  altitude  8  chains  50  links. 

24.  Sides  60  ft.,  80  ft,,  90  ft. 

25.  Sides  100  ft.,  110  ft.,  120  ft. 


MENSURATION.  285 


299.  The  area  of  any  surface  figure  bounded  by  straight 
lines  can  be  found  by  dividing  the  figure  into  triangles,  com- 
puting the  areas  of  these  triangles,  and  talcing  the  sum  of 
these  areas. 

300.  To  find  the  area  of  a  circle. 

Multiply  the  square  of  the  radius  by  3.1416.     (See  page 
168.) 

Find  the  area  of  a  circle,  having  given  : 

26.  Radius  14  ft.  28.    Diameter  32  yds. 

27.  Radius  7  yds.  29.    Diameter  40  ft. 

30.    Diameter  100  rds. 

301.  To  find  the  volume  of  a  rectangular  solid. 

Multiply  the  area  of  the  base  by  the  altitude ;  that  is,  take 
the  product  of  its  three  dimensions.     (See  page  174.) 

Find  the  volume  of: 

31.  A  cube  whose  edge  is  3  in. 

32.  A  cube  whose  edge  is  14  in. 

33.  A  cube  whose  edge  is  2  ft. 

34.  A  rectangular  solid  10"  X  8"  X  6". 

35.  A  rectangular  solid  6'  x  5'  X  4'. 

36.  A  rectangular  solid  4'  8"  X  3'  10"  X  3'  6". 

37.  If  a  cellar  which  measures  32f  X  28'  is  flooded  to  a 

depth  of  4  in.,  what  is  the  weight  of  the  water, 
allowing  1  cu.  ft.  of  water  to  weigh  1000  oz.? 

302.  To  find  the  number  of  gallons  that  a  cistern  of  given 
dimensions  will  hold, 

Find  the  number  of  cubic  inches  in  the  cistern,  and  divide 
this  number  by  231. 


286  MENSURATION. 


Ex.    If  a  rectangular  cistern  is  4'  X  3f  4"  X  3',  how  many 
gallons  of  water  will  it  hold  ? 

4  ft.  =  48  in.        3  ft.  4  in.  =  40  in.         3  ft.  =  36  in. 
Since  a  gallon  is  231  cu.  in.,  the  number  of  gallons  is 
48  x  40  x  36  _  9Qq  9 

231  299.2  gals.  Ans. 

303.  Since  a  cubic  foot  contains  -y^  gals.  =  7.48  gals., 
we  may  find  the  number  of  gallons  of  water  a  cistern  will 
hold  as  follows : 

Express  the  dimensions  in  feet,  and  multiply  the  continued 
product  of  these  dimensions  by  7J,  and  take  from  the  result 
io/1%  of  it. 

304.  To  find  the  number  of  barrels  of  water  a  cistern  will  hold. 
Divide  the  number  of  cubic  feet  the  cistern  contains  by 

4.21. 

Ex.  Find  the  number  of  gallons  in  a  cistern  6  ft.  square 
and  4  ft.  deep. 

6x6x4=    144 

7} 

1080 

Jofl%    = 5.4 

1074.6  gals.  Am. 

Ex.  Find  the  number  of  gallons  that  a  round  cistern  7  ft 

in  diameter  and  7  ft.  deep  will  hold. 
Area  of  base  =  3.1416  X  3.5  X  3.5  =         38.4846  sq.  ft. 

7 

269.3922  cu.  ft. 
7.5 


2020.44150 
£ofl%=       10.10 

2010.  gals.  Am. 


MENSURATION.  287 


38.  How  many  gallons  of  water  will  a  cistern  hold  that  is 

5J  ft.  long,  3f  ft.  wide,  and  4  ft.  deep? 

39.  How  many  barrels  of  water  will  a  cistern  hold  that  is 

13  ft.  long,  8  ft.  wide,  and  7  ft.  deep  ? 

40.  How  many  barrels  of  water  will  a  cistern  hold  that  is 

12  ft.  long,  9  ft.  wide,  and  6  ft.  deep  ? 

41.  Find  the  number  of  gallons  that  a  round  cistern  will 

hold,  8  ft.  in  diameter  and  7  ft.  deep. 

42.  Find  the  number  of  barrels  in  a  round  cistern  21  ft.  in 

diameter  and  10  ft.  deep. 

305.  To  find  the  number  of  bushels  of  grain  in  a  bin. 

A  bushel  =  2150.42  cu.  in.,  and  0.8  of  this  equals 
1720.336  cu.  in.  If  we  add  to  1720  i  of  1%  of  1720, 
rejecting  the  decimals,  we  obtain  1728  cu.  in. 

Hence,  take  0.8  of  the  number  of  cubic  feet  in  the  bin,  and 
add  to  the  result  i  ofl%  of  it. 

Ex.  Find  the  number  of  bushels  in  a  bin  12  ft.  long,  8  ft. 
wide,  and  5  ft.  high. 

12x8x5=  480 
0.8 

384.0 
}ofl%  -  1.92 

385.92  bu.  Ans. 

43.  Find  the  number  of  bushels  in  a  bin  20  ft.  long,  6  ft. 

wide,  and  4  ft.  high. 

44.  Find  the  number  of  bushels  in  a  bin  8}  ft.  long,  5J  ft. 

wide,  and  4  ft.  high. 

45.  Find  the  number  of  bushels  in  a  bin  8  ft.  long,  6f  ft. 

wide,  and  4}  ft.  high. 

306.  To  express  in  cubic  feet  a  given  number  of  bushels. 

To  the  number  of  bushels  add  }  of  the  number,  and  sub- 
tract from  the  sum  %of\°//)  of  it. 


288  MENSURATION. 


Ex.  To  find  the  number  of  cubic  feet  required  for  1200  bu. 

4)1200 
300 
1500 
Jofl%  =  7.5 

1492.5  cu.  ft.  Ans. 

46.  How  many  cubic  feet  in  a  bin  that  will  hold  400  bu.  ? 

47.  How  many  cubic  feet  in  a  bin  that  will  hold  372  bu.  ? 

48.  How  many  cubic  feet  in  a  bin  that  will  hold  1326  bu.  ? 

307,  To  find  the  number  of  bushels  in  a  load  of  charcoal. 
Multiply  the  continued  product  of  the  length,  width,  and 

height,  expressed  infect,  by  0.8  and  add  to  the  result  1  of 
1%  of  it. 

49.  How  many  bushels  of  charcoal  in  a  load  8  ft.  long,  4 

ft.  wide,  and  6  ft.  high  ? 

50.  Find  the  number  of  bushels  in  a  load  of  charcoal  that 

is  8  ft.  long,  41  ft.  wide,  and  6  ft.  high. 

308,  To  measure  wood. 

Find  the  product  of  the  length,  width,  and  height,  ex- 
pressed in  feet,  and  divide  this  product  by  8x4x4.  The 
result  is  the  number  of  cords. 

Ex.   Find  the  number  of  cords  in  a  pile  of  wood  24  ft.  long, 
4  ft.  wide,  and  6  ft.  high. 

^ 


2 

51.  What  part  of  a  cord  does  a  load  of  wood  contain  which 

is  8  ft.  long,  4  ft.  wide,  3J  ft.  high? 

52.  What  part  of  a  cord  does  a  load  of  wood  contain  which 

is  8  ft.  long,  3  ft.  8  in.  high,  if  the  average  length 
of  the  sticks  is  only  3  ft.  8  in.  ? 


MENSURATION.  289 


53.  Find  the  number  of  cords  in  a  pile  of  wood  120  ft. 

long,  4  ft.  wide,  and  6  ft.  high. 

54.  How  much  should  be  paid  for  a  pile  of  4-foot  wood, 

100  ft.  long,  and  averaging  5  ft.  high,  at  $5  a  cord? 

309.  To  measure  coal. 

A  long  ton  of  anthracite  coal  measures  about  37  cu.ft. 
A  long  ton  of  soft  coal  measures  about  48  cu.  ft.  A  bushel 
of  hard  coal  weighs  about  75  Ibs.  A  bushel  of  soft  coal 
weighs  about  60  Ibs. 

55.  How  many  long  tons  of  hard  coal  will  a  rectangular 

bin  hold  9  ft.  long,  6  ft.  6  in.  wide,  and  6  ft.  high  ? 

56.  How  many  long  tons  of  hard  coal  can  be  put  into  a 

rectangular  bin  8  ft.  long,  7  ft.  wide,  and  6  ft.  high  ? 

57.  How  many  long  tons  of  soft  coal  can  be  put  into  a  rec- 

tangular bin  12  ft.  long,  9  ft.  wide,  and  7  ft.  high? 

310.  To  measure  sand,  gravel,  and  earth. 
A  cubic  yard  of  earth  is  called  a  load. 

58.  How  many  loads  are  there  in  a  rectangular  embank- 

ment 200  ft.  long,  15  ft.  wide,  and  10  ft,  high  ? 

59.  How  many  loads  in  an  embankment  150  ft.  long,  20 

ft.  wide,  and  5  ft.  high  ? 

311.  To  measure  brickwork. 

Brickwork  is  estimated  by  the  thousand,  reckoning  22 
bricks  laid  in  mortar  to  the  cubic  foot. 

60.  How  many  bricks  will  be  required  to  build  a  wall  84 

ft.  long,  32  ft.  high,  and  1  ft.  thick? 

61.  How  many  bricks  will  be  required  for  the  walls  of  a 

house  42  ft.  long,  32  ft.  wide,  and  21  ft.  high,  if  the 


290  MENSURATION. 


walls  are  1  ft.  thick,  and  there  are  deducted  2  doors 
7'  6ff  X  4r  each,  and  16  windows  5'  X  4'  each  ? 

NOTE.  In  finding  the  perimeter  of  the  building,  measure  the  exte- 
rior. 

312.  To  measure  stone  masonry. 

Stone  masonry  is  reckoned  by  the  cubic  foot,  or  by  the 
perch  of  25  cu.  ft. 

62.  How  many  cubic  feet  of  stone  masonry  in  the  foun- 

dation of  a  house  40r  X  30r,  if  the  foundation  is  to 
be  4  ft,  high  and  2  ft.  thick  ? 

63.  How  many  perches  of  stone  are  required  for  the  foun- 

dation of  a  building  100'  X  60',  if  the  foundation  is 
6  ft.  high  and  2£  ft.  thick  ? 

313.  To  measure  boards  and  dimension  lumber. 

Boards  one  inch  or  less  in  thickness  are  sold  by  the 
square  foot.  Boards  more  than  one  inch  in  thickness  and 
all  squared  lumber  are  sold  by  the  number  of  square  feet 
of  boards  one  inch  in  thickness  to  which  they  are  equivalent. 

Thus,  a  board  12  ft.  long,  1  ft.  wide,  and  1  in.  thick, 
contains  12  ft.  board  measure.  If  only  i  or  I  or  J  of  an 
inch  thick,  it  still  contains  12  ft.  board  measure;  but  if  li 
in.  thick,  it  contains  It  X  12  =  15  ft.  board  measure. 
Hence, 

Express  the  length  and  width  in  feet,  and  the  thickness  in 
inches.  The  product  of  these  three  numbers  will  be  the 
number  of  feet  board  measure. 

In  practice,  the  width  of  a  board,  unless  sawed  to  order,  is  reck- 
oned only  to  the  next  smaller  half-inch.  Thus,  a  width  of  llf  inches 
is  reckoned  11  inches ;  of  13f  or  13J  inches,  is  reckoned  13J  inches. 

Ex.  How  many  feet  in  a  2-inch  plank,  18  ft.  long  and 

14  in.  wide? 
14  in.  =  1J  ft.         2  X  1J  X  18  =  42  ft.  board  measure.  Am. 


MENSUPwATION.  291 


64.  How  many  feet  board  measure  in  8  planks,  4  in.  thick, 

18  ft.  long  and  16  in.  wide? 

65.  How  many  feet  board  measure  in  a  stick  of  timber 

1  ft.  square  and  20  ft.  long? 

66.  How  many  feet  board  measure  in  40  joists  10"  X  2" 

and  12ft.  long? 

314,  To  measure  round  logs.     Round  logs  are  sold  by  the 
amount  of  square  lumber  that  can  be  cut  from  them,  ac- 
cording to  calipers  now  in  use.     When  logs  do  not  exceed 
16  ft.  in  length,  the  length  and  the  diameter  of  the  small 
end  are  taken,  and  a  table  stamped  upon  the  calipers  gives 
the  number  of  feet  board  measure.      This  table  may  be 
calculated  as  follows : 

Express  the  diameter  in  inches,  subtract  twice  the  diameter 

from  the  square  of  the  diameter,  and  f  ^  of  the  remainder 

will  be  the  number  of  feet  board  measure  in  a  log  10  ft.  long. 

The  formula  is  %$(d2  —  2c?),  in  which  d  stands  for  the 

diameter  of  the  log  in  inches. 

Ex.    Find  the  number  of  feet  board  measure  in  a  log  16  ft. 
long  and  20  inches  in  diameter. 
202  -  2  x  20  =  400  -  40  =  360. 
ft  of  360  =  189. 
jf  of  189  =  302.4  ft.  board  measure.  Ans. 

By  this  rule  find  the  number  of  feet  board  measure  in : 

67.  A  log  12  ft.  long  and  16  in.  in  diameter. 

68.  A  log  13  ft.  long  and  12  in.  in  diameter. 

69.  A  log  14  ft.  long  and  20  in.  in  diameter. 

70.  A  log  15  ft.  long  and  15  in.  in  diameter. 

315.  Oak  and  other  heavy  timber. 

Large  heavy  timber  of  hard  wood  is  generally  sold  by  the 
ton,  signifying  50  cu.ft.  or  600 /£.  board  measure. 


292  MENSURATION. 


316.  To  find  the  contents  of  a  cask. 

Subtract  the  diameter  of  one  of  the  heads  from  the  bung 
diameter  expressed  in  inches,  and  multiply  the  difference  by 
0.65 ;  to  the  product  add  the  head  diameter,  and  this  will 
give  the  mean  diameter. 

Square  the  mean  diameter  and  multiply  it  by  the  length 
in  inches.  Divide  this  product  by  294.  The  quotient  is  the 
number  of  gallons  the  cask  will  hold. 

71.  Find  the  number  of  gallons  contained  in  a  cask  of 

which  the  bung  diameter  is  24  in.,  head  diameter 
20  in.,  and  the  length  36  in. 

72.  Find  the  number  of  gallons  contained  in  a  cask  of 

which  the  bung  diameter  is  30  in.,  head  diameter 
26  in.,  and  the  length  38  in. 

317.  To  find  the  volume  of  an  irregular  body. 

Immerse  the  body  in  a  vessel  full  of  water.  JRemove  the 
body  and  calculate  the  amount  of  water  displaced. 

318.  To  find  the  surface  of  a  sphere. 
Multiply  the  square  of  the  diameter  by  3.1416. 

73.  How  many  square  inches  on  the  surface  of  a  ball  4  in. 

in  diameter? 

74.  How  many  square  inches  on  the  surface  of  a  globe  18 

in.  in  diameter? 

319.  To  find  the  volume  of  a  sphere. 

Multiply  the  cube  of  the  diameter  by  0.5236  (that  is,  £  of 
3.1416). 
75     Find  the  volume  of  a  globe  2  ft.  in  diameter. 


CHAPTER  XV. 

Ex.  164. 
MISCELLANEOUS  PROBLEMS. 

1 .  Fifteen   men   and   eight   boys  together  earn  $  342  a 

week.  If  a  boy's  pay  is  half  a  man's  pay,  what  are 
the  daily  wages  of  a  man,  and  also  of  a  boy  ? 

2.  A  man  divides  $1622.50  among  four  persons  so  that 

the  first  has  $40  more  than  the  second,  the  second 
$60  more  than  the  third,  and  the  third  $87.50  more 
than  the  fourth.  Find  the  part  of  the  fourth. 

3.  A  family  of  six  persons  makes  $8.75  a  day,  and  works 

304  days  in  the  year.  At  the  end  of  the  year  each 
member  of  the  family  puts  $80  in  a  savings  bank. 
Find  the  daily  expense  of  the  family. 

4.  A  man  bought  5.5  yds.  of  cloth  for  $35.     In  having 

a  suit  made  from  it,  he  found  that  he  lacked  1.75 
yds.,  which  he  procured  at  the  price  per  yard  of 
his  first  purchase.  What  is  the  cost  of  the  suit  if 
the  trimmings  cost  $6.50  and  the  making  $15? 

5.  A  man  has  76.25  yds.  of  linen,  worth  44  cts.  a  yard, 

made  into  shirts.  It  takes  3.05  yds.  for  a  shirt,  and 
the  price  for  making  is  50  cts.  a  shirt.  Find  the 
cost  of  a  shirt,  and  the  number  he  has  made. 

6.  A  man's  expenses  from  the  first  of  January  to  the  end 

of  October  17  are  $1845.50.  How  much  must  he 
diminish  his  daily  expense  in  order  that  the  total 
expense  for  the  year  shall  not  exceed  $2200? 


294  MISCELLANEOUS   PROBLEMS. 

7.  A   quart  contains  1600  beans  of  average  size,  and  a 

field  is  planted  with  22  rows  of  800  hills  each,  with 
6  beans  in  a  hill.  The  increase  is  tenfold.  What 
is  the  value  of  the  crop  at  $3  a  bushel?  (There 
are  32  quarts  in  a  bushel.) 

8.  For  making  25  gallons  of  ordinary  beer  60  pounds  of 

barley  and  0.5  of  a  pound  of  hops  are  needed.  If 
the  barley  costs  $1.50  for  60  pounds,  and  the  hops 
cost  18  cents  a  pound,  what  is  the  profit  to  the 
brewer  on  a  cask  of  42  gallons  if  he  sells  it  for  $5 
and  reckons  his  labor  $1.50? 

9.  A  person  receives  his  income   quarterly.      The  first 

quarter  he  receives  $533.25,  the  second  $1535.20, 
the  third  $856.44,  the  fourth  $  725.19.  His  expenses 
for  these  quarters  are  respectively  $686.60,  $734.25, 
$589.15,  $849.65.  How  much  does  he  save  for  the 
year? 

10.  A  hen  lays  on  an  average  120  eggs  a  year  worth  24 

cents  a  dozen.  She  eats  a  quart  of  barley  every  5 
days.  The  barley  is  worth  56  cents  a  bushel  (32 
quarts).  What  is  the  annual  profit  from  this  hen  ? 

11.  A  square  garden  measuring  on  each  side  40.50  yards 

is  enclosed  by  three  lines  of  galvanized  iron  wire. 
Eight  yards  of  this  wire  weigh  a  pound,  and  it  is 
worth  7.5  cents  per  pound.  What  is  the  cost  of  the 
wire  ? 

12.  A  family  composed  of  five  persons  consumes  daily  one 

pound  of  stale  bread  for  each  person,  or  1.15  pounds 
of  fresh  bread.  If  bread  is  worth  5  cents  a  pound, 
find  the  annual  saving  which  this  family  will  make 
if  it  eats  stale  bread  altogether. 


MISCELLANEOUS   PROBLEMS.  295 

13.  Tfc  is  estimated  that  in   France  240,000  women   and 

girls  are  employed  in  making  lace.  The  annual 
production  has  a  value  of  $13,000,000,  and  the 
value  of  the  raw  material  is  0.27  of  the  value  of 
the  lace.  Find  the  average  daily  wages  of  these 
women  and  girls,  supposing  that  each  works  240 
days  in  the  year. 

14.  The  salt  water  which  is  obtained  from  the  bottom  of  a 

mine  of  rock  salt  contains  0.09  of  its  weight  of  pure 
salt.  What  weight  of  salt  water  is  it  necessary  to 
evaporate  in  order  to  obtain  4734  pounds  of  salt  T 

15.  The  weight  of  ashes  from  the  burning  of  oak  wood  is 

0.03  of  the  weight  of  the  wood,  and  the  weight  of 
carbonate  of  potash  contained  in  the  ashes  is  0.065 
of  the  weight  of  the  ashes.  Find  the  weight  of 
carbonate  of  potash  from  1170  pounds  of  wood. 

16.  The  weight  of  sugar  from  the  sugar  beet  is  nearly  0.06 

of  the  weight  of  the  beet.  If  an  acre  produces  30,000 
pounds  of  beets  that  are  sold  at  the  rate  of  $2  a 
thousand  pounds,  how  many  acres  of  land  is  it 
necessary  to  sow  to  furnish  beets  to  a  sugar  factory 
which  produces  150,000  pounds  of  sugar  a  year,  and 
what  will  be  the  value  of  the  crop  obtained  ? 

17.  If  a  workman  has  taken   every  day  for  the  last   12 

years  two  glasses  of  beer  at  5  cents  a  glass,  how 
much  could  he  have  saved  if  he  had  not  indulged 
this  habit,  reckoning  365  days  each  year? 

18.  A  woman  has  three  children.     She  pays  for  each  $15 

a  year  for  having  their  clothes  made,  $1.50  a  month 
for  mending,  and  $0.35  a  week  for  washing.  How 
much  could  she  save  in  a  year  if  she  knew  how  to 
wash,  make  clothes,  and  mend  ? 


296  MISCELLANEOUS   PROBLEMS. 

19.  A  sheep  raiser  shears  his  sheep  at  an  expense  of  11  cts. 

a  head.  The  sheep  average  8  Ibs.  of  wool  which 
he  sells  for  23  cts.  a  pound.  He  finds  that  his  net 
profit  after  paying  for  the  shearing  is  $1297.50. 
How  many  sheep  has  he  ? 

COMMON  FRACTIONS. 

20.  Find  the  prime  factors  of  41,580. 

21.  Find  the  G.C.M.  of  144,  126,  108. 

22.  Find  the  L.C.M.  of  18,  90,  60,  24. 

23.  Find  the  L.C.M.  of  14,  35,  343. 

24.  At  16£  cts.  a  yard,  what  will  3£  yds.  of  cloth  cost? 

25.  A  man  has  376|-  quarts  of  berries,  which  he  wishes  to 

put  into  boxes  holding  2J-  qts.  each.  How  many 
boxes  will  be  required,  and  what  part  of  a  box  will 
be  left  over? 

26.  If  a  man  earns  $2f  a  day,  how  many  days  will  it  take 

him  to  earn  $100? 

27.  A  lady  has  37-J-  qts.  of  berries  to  can.     If  each  can 

holds  2f  qts.,  how  many  cans  of  berries  will  she 
have,  and  what  part  of  another  can  will  there  be 
over? 

28.  If  a  man  walks  4|-  miles  an  hour,  how  many  hours 

will  it  take  him  to  walk  40f  miles  ? 

29.  Some  boys  wanted  a  long  rope  to  use  on  the  ice.   They 

made  the  rope  by  taking  off  their  sled-ropes  and 
tying  them  together.  The  first  sled-rope  was  2f 
yds.  long,  the  second  3^-  yds.,  the  third  2£  yds.,  the 
fourth  5f  yds.,  and  the  fifth  3^  yds.  If  the  whole 
length  was  shortened  1-|-  yds.  by  the  knots,  from 


MISCELLANEOUS  PROBLEMS.  297 

tying  the  sled-ropes  together,  how  long  was  the 
rope? 

30.  A  lady  bought  3|-  yds.  of  cotton  cloth,  4J-  yds.  of 

calico,  16|  yds.  of  flannel,  and  12-J-  yds.  of  ging- 
ham. How  many  yards  did  she  buy  in  all  ? 

31.  A  boy  went  to  a  store  with  $5.75  in  his  purse.     He 

bought  3^  Ibs.  of  butter  at  28  cts.  a  pound,  13^  Ibs. 
of  sugar  at  11  cts.  a  pound,  and  1^-  Ibs.  of  coffee  at 
35  cts.  a  pound.  How  much  money  did  he  have 
left? 

32.  Four  boys  went  fishing,  and  caught  40  trout ;  the  first 

caught  -§-  of  the  whole,  the  second  -J-,  and  the  third 
•J.  How  many  did  the  fourth  boy  catch  ? 

33.  George  has  his  choice  to  be  one  of  3  boys  to  receive  8 

oranges,  or  one  of  4  boys  to  receive  11  oranges. 
Which  shall  he  choose? 

34.  Five   girls  pick  blueberries  together ;    the  first  picks 

7|  qts.,  the  second  5^  qts.,  the  third  12|  qts.,  the 
fourth  8%  qts.,  and  the  fifth  3£  qts.  How  much 
will  they  all  together  get  for  their  berries,  at  12^ 
cts.  a  quart? 

35.  A  farmer  puts  the  following  lots  of  apples  into  6  bins: 

namely,  6f  bu.,  18£  bu.,  25£  bu.,  19f  bu,  143|  bu., 
976^  bu.,  25£  bu.  How  many  bushels  will  there  be 
for  each  bin  ? 

COMPOUND  QUANTITIES. 

36.  How  many  rods  are  there  in  4379  ft.  ? 

37.  Reduce  9,627,834  ft.  to  yards,  rods,  etc. 

38.  Reduce  96,284  sq.  in.  to  square  feet. 


298  MISCELLANEOUS  PROBLEMS. 

39.  Reduce  15  sq.  rds.  3  sq.  yds.  18  sq.  ft.  3  sq.  in.  to 

square  inches. 

40.  What  will  1000  sq.  ft.  of  land  cost  at  $67  an  acre? 

41.  What  will  20  sq.  yds.  of  land  cost  at  75  cts.  a  square 

foot? 

42.  How  much  less  will  15  acres  of  land  cost,  at  $16  an 

acre,  than  96,342.42  sq.  ft.  at  5  cts.  a  foot? 

43.  How  many  acres  in  a  rectangular  piece  of  land  963$ 

ft.  long  and  3840  ft.  wide? 

44.  A  pile  of  four-foot  wood  is  4  ft.  high  and  75  ft.  long. 

How  many  cords  of  wood  are  there  in  the  pile  ? 

45.  In  a  woodshed  there  is  a  pile  of  wood  12  ft.  long  and 

10  ft.  high.     If  the  sticks  average  a  foot  in  length, 
what  part  of  a  cord  is  there  in  the  pile? 

46.  What  will  7  bu.  3  pks.  of  blueberries  bring  at  9  cts.  a 

quart  ? 

47.  How  many  gallons  of  milk,  at  8  cts.  a  quart,  can  be 

bought  for  $7.37? 

48.  How  many  quarts  of  water  will  a  tin  box  hold  that  is 

13  in.  long,  6  in.  wide,  and  7  in.  deep? 

49.  The  total  net  weight  of  several  loads  of  hay  is  63,782 

Ibs.     How  many  tons  in  all  the  loads  of  hay  ? 

50.  If  an  ounce  of  candy  is  worth  5  cts.,  what  will  5  Ibs. 

cost  at  the  same  rate? 

51.  Reduce  9  dys.  5  hrs.  16  min.  to  seconds. 

52.  Reduce  948,741  min.  to  higher  denominations. 

53.  How  many  weeks  between  Jan.  1  and  Nov.  1? 

54.  A  boy  has  10  mi.  to  go.     After  he  has  gone  6  mi.  48 

rds.  12  ft.,  how  much  of  his  journey  has  he  still  to 
go? 


MISCELLANEOUS   PROBLEMS.  299 

55.  A  lady  bought  4  remnants  of  cloth;   the  first  con- 

tained 9J  yds.,  the  second  4  yds.  11  in.,  the  third 
6J  yds.,  and  the  fourth  5-}  yds.  How  much  cloth 
did  she  buy  in  all  ? 

56.  A  certain  basket  holds  1  bu.  3  pks.  7  qts.     A  farmer 

raises  enough  of  yellow-eyed  beans  to  fill  this  bas- 
ket 7  times.  How  many  bushels  does  he  raise  ? 

57.  A  farmer  cuts  26  loads  of  hay,  which  average  1  t. 

436  Ibs.     How  many  tons  does  he  cut  in  all  ? 

58.  What  is  ^  of  9  mi.  5  rds.  13  ft.  ? 

59.  Three  men  in  company  buy  175  t.  19  cwt.  36  Ibs.  of 

hay.     What  is  each  man's  share? 

GO.    Seven  boys  together  pick  4  bu.  3  pks.  7  qts.  of  berries. 
What  is  each  boy's  share  ? 


61.  Bought  9  Ibs.  of  sugar  at  13  cts.  a  pound,  18  yds.  of 

cloth  at  33  cts.  a  yard,  4  doz.  eggs  at  29  cts.  a 
dozen;  and  5  Ibs.  of  butter  at  32  cts.  a  pound. 
What  change  should  I  receive  from  a  ten-dollar  bill 
given  in  payment? 

62.  How  many  quarts  of  berries,  at  12  cts.  a  quart,  will 

it  take  to  pay  for  8  yds.  of  cloth,  at  16  J  cts.  a  yard? 

63.  A    basket  of  peaches    is    half   a    bushel ;    how  many 

bushels  are  there  in  250  car-loads  of  500  baskets 
each  ? 

64.  A  fast  railway  train  in  England  went  186  mi.  240  rds. 

in  3  hrs.     What  was  the  rate  per  hour  ? 

65.  Tf  a  man  could  proceed  to  the  moon  at  the  same  rate 

per  hour  as  the  train  went  in  example  64,  how 
many  hours  would  it  take  him,  reckoning  the  dis- 
tance 239,000  miles? 


300  MISCELLANEOUS   PROBLEMS. 

66.  In  one  bin  there  are  23  bu.  2.48  pks.  of  wheat,  and  in 

another  141  bu.  2  pks.  If  J  of  the  wheat  in  the  first 
bin  is  put  into  the  second,  how  much  wheat  will 
there  be  in  the  second  bin  ? 

67.  A  load  of  four-foot  wood  is  3}  ft.  high  and  7  ft.  long. 

What  is  it  worth  at  the  rate  of  $6.40  a  cord? 

68.  In  one  field  there  are  17J  A.,  in  a  second  there  are 

49  sq.  rds.,  and  a  third  field  is  1740  ft.  long  and 
927  ft.  wide.  What  is  the  area  of  the  three  fields 
together  ? 

69.  A  bin  contains  164  bu.  3  pks.  2  qts.  of  oats.     How 

long  will  these  oats  last  if  there  are  taken  out  3 
qts.  of  oats  three  times  a  day  ? 

70.  From  a  barrel  containing  27f  gals,  of  oil,  3  qts.  a  day 

were  taken  out  for  3  weeks.  How  many  gallons 
were  left  in  the  barrel  at  the  end  of  that  time  ? 

71.  If  from  a  barrel  of  oil  holding  27  gals.  2  qts.  1  pt. 

there  is  drawn  out  a  can  full,  holding  1  gal.  2  qts. 
1  pt.,  every  day,  how  many  days  will  the  oil  last? 

72.  Reduce  £  of  -f-  of  {^  of  a  mile  to  rods. 

73.  Reduce  $  of  -j-  of  3£  in.  to  the  fraction  of  a  yard. 

SPECIAL  PROBLEMS. 

If  a  man  can  do  a  piece  of  work  in  5  dys.,  in  one  day  he  can  do 
J  of  the  work  ;  and  if  another  man  can  do  the  same  work  in  4  dys., 
in  one  day  he  can  do  J  of  it. 

Therefore,  both  men  together  can  do  £  +  J  =  ^  in  one  day. 

Hence  they  will  do  •£$  in  ^  of  a  day,  and  therefore  the  whole 
work  in  *£•  days,  that  is,  in  2f  days. 

74.  If  A  can  do  a  piece  of  work  in  4  dys.,  B  in  5  dys., 

and  0  in  7  dys.,  in  how  many  days  will  they  do  it, 
all  working  together  ? 


MISCELLANEOUS   PROBLEMS.  301 

75.  A  can  do  a  piece  of  work  in  2  hrs.,  B  in  2-J-  hrs.,  and 

C  in  3^  hrs.  How  much  of  the  work  can  they  do 
in  20  min.,  all  working  together? 

76.  If  A  and  B  can  do  a  piece  of  work  in  18  dys.,  A  and 

C  in  12  dys.,  and  B  and  C  in  9  dys.,  find  the  num- 
ber of  days  that  it  will  take  them,  all  working 
together. 

77.  A  can  do  a  piece  of  work  in  6  dys.,  B  in  8  dys.,  and 

G  in  10  dys.  How  much  of  it  can  they  do  in  2  dys. 
together  ? 

78.  A  cistern  can  be  filled  by  means  of  a  water-pipe  in  30 

min.,  and  can  be  emptied  by  a  waste-pipe  in  20  min. 
If  the  cistern  is  full,  and  both  pipes  are  open,  in 
what  time  will  it  be  emptied  ? 

79.  From  Paris  to  Berlin  by  railway  it  is  1308km.     A  kilo- 

meter is  1093.63  yds.  Express  the  distance  between 
Paris  and  Berlin  in  miles  and  yards. 

80.  Mercury  revolves  around  the  sun  in  87.9692580  dys. 

Express  the  period  of  revolution  in  days,  hours,  min- 
utes, and  seconds. 

81.  The  Roman  foot  was  0.97075  of  our  foot.     The  Greek 

foot  was  -||  of  the  Roman  foot.  Find  the  length  in 
inches  of  the  Greek  foot. 

82.  The  radius  of  a  circle  is  0.1591549  of  its  circumfer- 

ence, which  contains  360°.  Find  the  angle  at  the 
centre  whose  arc  is  equal  to  the  radius. 

83.  Find  the  L.C.M.  of  all  the  multiples  of  3,  from  6  to 

27,  inclusive. 

84.  Arrange  -|,  -|-J,  and  |-jj-  in  order  of  magnitude. 

85.  Subtract  the  sum  of  -f,  £,  f ,  ||,  ^  from  5. 


302  MISCELLANEOUS   PROBLEMS. 

86.  Find  the  decimal  which,  when  added  to  the  difference 

of  -^-g-  and  0.002775,  produces  the  square  of  0.215. 

87.  A,  at  the  rate  of  4*-  miles  an  hour,  walks  a  certain 

distance  in  3^  hrs.  In  what  time  will  B  walk  the 
same  distance  at  the  rate  of  |  of  5J-  miles  an  hour? 

PERCENTAGE. 

88.  A  house  wrorth  $15,000  sustains  injury  from  fire  to 

the  amount  of  $3840.  What  is  the  rate  per  cent 
of  loss? 

89.  A  and  B  have  each  $350;    A  spends  1G%    and  B 

spends  20%.  A's  expenditure  is  what  per  cent  of 
B's? 

90.  A  gentleman  having  a  court  20  ft.  by  40  ft.  enlarged 

it  10%  in  each  dimension.  Find  the  per  cent  of 
increase  in  area. 

91.  A  young  man  buys  a  farm  for  $5200,  which  sum  is 

30%  more  than  a  legacy  received  from  his  grand- 
father. Required  the  amount  of  the  legacy. 

92.  A  lady  gave  to  her  daughter  25%  and  to  her  son  20% 

of  her  estate.  The  difference  between  the  shares  of 
the  son  and  daughter  was  $1500.  What  is  the 
value  of  the  estate? 

93.  If  a  quart  of  Jersey  milk  is  worth  10  cts.,  and  pro- 

duces 1  gi.  of  cream  worth  25  cts.  a  pint,  what  per 
cent  of  the  value  of  the  milk  is  the  value  of  the 
cream  ? 

94.  A  farmer  raised  360  bu.  of  potatoes,  and  the  crop  was 

2400%  of  the  seed.  How  many  bushels  did  he 
plant  ? 

95.  A  man  received  from  a  bankrupt  $937.50,  which  was 

of  the  sum  due.     What  was  his  loss? 


MISCELLANEOUS   PROBLEMS.  303 

96.  What  per  cent  of  f  is  \  ? 

97.  If  200%  of  a  number  is  \<J0  of  70,  what  is  the  num- 

ber? 


98.  A  grocer  sold  10%    of  his  stock  of  sugar,  and  then 

25%  of  the  remainder,  after  which  he  had  3  t. 
1560  Ibs.  How  much  sugar  had  he  at  first? 

99.  A  man  lost  37i%   of  his  money.     He  then  earned 

$50,  and  had  125%  of  what  he  had  at  first.  How 
much  did  he  have  at  first  ? 

100.  A   merchant  bought  a  cask  of  molasses  from  which 

20%  of  the  molasses  had  been  drawn.  He  sold 
30  \  gals.,  and  then  the  cask  was  one-quarter  full. 
Find  the  capacity  of  the  cask  in  gallons. 

101.  What  per  cent  of  a  common  year  is  the  time  from 

July  1  to  November  23,  both  days  included  ? 

102.  A  horse  and  chaise  together  are  valued  at  $225  ;  the 

horse  is  worth  25%  more  than  the  chaise.  Find 
the  value  of  the  horse. 

103.  A  man  owning  30%  of  a  mine  sold  50%  of  his  share 

for  $3000.     What  was  the  value  of  the  mine  ? 

104.  For  what  price  per  pair  must  shoes  be  sold  to  gain 

25%,  if  15%  is  lost  when  they  are  sold  at  $1.275 
per  pair  ? 

105.  If  -£•  of  goods  valued  at  $1500  are  sold  at  a  loss  of 

10%,  what  must  the  remainder  bring  to  gain  20% 
on  the  whole  ? 

106.  A  fruit  dealer  bought  200  apples  at  the  rate  of  4  for 

a  cent,  and  200  at  5  for  a  cent.  He  sold  them  all 
at  5  for  3  cents.  What  per  cent  did  he  gain  on 
his  investment? 


304  MISCELLANEOUS   PROBLEMS. 

107.  If  75%  of  the  price  of  a  bushel  of  corn  is  50%  of  the 

price  of  a  bushel  of  wheat,  how  many  bushels  of 
corn  can  be  bought  for  $24  when  wheat  is  worth 
$1.20  a  bushel? 

108.  A  horse  dealer  sold  a  horse  for  $90,  and  lost  25%  of 

the  cost  of  the  horse.  He  sold  another  horse  at 
an  advance  of  20%  on  the  cost,  and  gained  as  much 
as  he  lost  on  the  first  horse.  What  was  the  selling 
price  of  the  second  horse  ? 

109.  If  20  men  can  build  a  wall  in  9  dys.,  what  per  cent 

of  the  number  of  men  could  build  the  wall  in  12 
dys.? 

110.  If  7%  of  a  ton  of  butter  costs  $42,  what  per  cent  of 

a  ton  can  be  bought  for  $57  ? 

111.  Five  hundred  barrels  of  flour  were  sold  for  $4125,  at 

a  profit  of  10%.     Find  the  cost  per  barrel. 

112.  An  agent  makes  20%  by  selling  a  book  for  72  cts. 

If  he  had  sold  it  for  $1,  what  per  cent  would  he 
have  made  ? 

113.  A  merchant  bought  from  a  shoe  dealer  12  cases  of 

shoes,  each  containing  60  pairs,  at  87-^-  cts.  per 
pair,  and  sold  the  whole  for  $756.  Find  his  gain 
per  cent. 

114.  If  196  sq.  rds.  are  40%  of  the  area  of  a  field  30  rds. 

in  length,  what  is  the  width  of  the  field  ? 

115.  When  brooms  are  $5.50  a  dozen,  what  will  be  paid 

for  18|-  gross,  if  a  discount  of  10%  is  allowed  on 
the  bill  for  cash  ? 

116.  A  grocer  bought,  at  60  cts.  per  gallon,  16  hhds.  of 

molasses  of  63  gals,  each,  and  sold  it  at  a  profit 
of  $120.96.  What  was  his  gain  per  cent? 


MISCELLANEOUS   PROBLEMS.  305 

117.  A  merchant  in  his  first  year  of  business  increased  his 

capital  40%,  and  increased  his  capital  the  second 
year  30%.  He  lost  33|-%  of  his  capital  the  third 
year,  and  had  $18,200  left.  What  was  his  capital 
at  first? 

118.  A  contractor  engaged  to  build  a  railroad  at  $31,200 

a  mile.  The  work  actually  cost  $90  per  rod. 
What  was  his  gain  per  cent  ? 

119.  A  merchant  sold  goods  at  25%  discount  and  4%  off 

from  the  selling  price  for  cash.  What  was  the 
whole  per  cent  discount? 

120.  At    \\°Jo   commission  an  agent  receives   $97.29  for 

selling  goods.     Find  the  amount  of  the  sale. 

121.  A  merchant  sent  $30,750  to  his  agent  in  New  Orleans, 

for  the  purchase  of  cotton.  Find  the  sum  spent  for 
cotton,  if  the  agent  charges  2J%  commission  for 
buying. 

122.  Find  the  sum  paid  for  insurance,  at  -£%,  on  a  house 

worth  $8000,  and  at  f  %  on  furniture  worth  $2000, 
if  the  insurance  is  on  -£  of  the  value  of  the  property 
insured. 


123.  A  sea  captain  paid  $345,  at  IY%,  for  insuring  f  of 

the  value  of  a  ship.     Find  the  value  of  the  ship. 

124.  A  town  has  to  raise  $192,000  for  expenses.     If  4% 

is  allowed  for  collecting,  how  much  money  must  be 
raised  ? 

125.  A  merchant  sends  $24,600  to  his  agent  at  St.  Louis, 

for  the  purchase  of  flour  at  $5  a  barrel.  How 
many  barrels  can  be  bought  if  the  agent  charges 
commission  for  buying? 


306  MISCELLANEOUS    PROBLEMS. 

126.  A  paper-mill   worth   $30,000   was    insured    for   an 

annual  premium  of  If  %  on  90%  of  its  value.  In 
the  second  year  it  was  injured  by  fire  to  the  amount 
of  $1780.  How  much  did  the  mill  owner  save  by 
insuring  ? 

127.  A  city  voted  a  tax  of  $74,500;  the  poll-tax  was  $1.25 

on  2000  polls ;  the  assessed  value  of  city  property 
was  $6,000,000.  What  was  the  tax  on  $1000? 

128  What  insurance  must  be  placed  upon  a  store  and  its 
contents,  valued  at  $20,085,  that  the  entire  value 
of  the  goods  and  store  and  of  a  premium  of  2£% 
may  be  recovered  in  case  of  loss  by  fire  ? 

129.  A  premium  of  $88.14  is  paid  upon  a  cargo  of  wheat 

insured  at  2f  %  on  £  of  its  value.  Find  the  num- 
ber of  bushels  shipped,  if  the  average  price  is  80 
cts.  a  bushel. 

130.  A  30%  duty  of  $5594.40  was  paid  on  252  watches. 

What  was  the  invoice  price  of  each  watch? 

131.  Eleven  and  one-half  yards  of  cloth  1-J  yds.  wide  are 

required  for  a  dress.  How  many  yards  must  be 
bought  if  the  shrinkage  in  sponging  is  10%  in 
length  and  8%  in  width? 

132.  If  30%   of  a  merchant's  sales  is  profit,  what  is  his 

gain  per  cent  ? 

133.  A  merchant  insured  a  ship  and  cargo  at  4|%.     If 

$  158,650  cover  both  property  and  premium,  what 
is  the  value  of  the  ship  and  cargo  ? 

134.  How  much  money  must  be  sent  to  purchase  10,000 

bbls.  of  sugar,  at  $8.50  per  barrel,  if  the  commis- 
sion for  buying  is  3%,  and  the  sum  prepaid  for 
freight  is  $315? 


MISCELLANEOUS   PROBLEMS.  307 

INTEKEST. 
Find  the  interest  of  : 

135.  $1000  for  2  yrs.  7  mos.  18  dys.,  at  6%. 

136.  $1496  for  7  mos.  21  dys.,  at  6%. 

137.  $582  for  1  yr.  7  mos.  15  dys.,  at  6%. 

138.  $168  for  1  yr.  5  mos.  12  dys.,  at  2f  %. 

139.  $548  for  7  mos.  18  dys.,  at  6|%. 

140.  $1272  from  July  12,  1880,  to  Feb.  24,  1882,  at  3J%. 

141.  $1975.30  for  60  dys.,  at  6%. 

142.  $1675  for  90  dys.,  at  6%. 

143.  $976  for  3  yrs.  6  mos.,  at  1%  a  month. 

Find  the  rate  per  cent  : 

144.  When  the  interest  on  $3000  for  3  yrs.  is  $630. 

145.  When  the  interest  on  $1500  for  2  yrs.  is  $172.50. 

146.  When  the  interest  on  $1278.50  for  3  yrs.  6  mos.  is 

$178.99. 

147.  When  a  sum  of  money  is  doubled  in  8  yrs. 

148.  When  $1758  amount  to  $1869.34  in  8  mos. 

Find  the  time  : 

149.  When  the  interest  on  $278.40,  at  7-i%,  is  $100.92. 

150.  When  $600,  at  3£%,  amount  to  $660. 


151.  When   the   interest   on   $78,    at    \\°/o    a  month,  is 

$28.08. 

152.  When  the  principal,  at  5%,  is  doubled. 


308  MISCELLANEOUS  PROBLEMS. 

Find  the  principal  that  will : 

153.  Produce  $424.94  interest  in  3  yrs.,  at 

154.  Produce  $285.60  interest,  at  7%,  in  1  yr.  8  mos.  12 

dys. 

155.  Produce  $81.37  interest,  at  3f  %,  in  2  yrs.  9  mos. 

18  dys. 

What  principal  will  amount  to  : 

156.  $88.80,  at  6%,  in  3  yrs.  4  mos. 

157.  $308.10,  at  5f%,  in  6  mos. 

158.  $570.475,  at  6%,  in  3  yrs.  4  mos.  6  dys. 

159.  $661.32,  at  \°f0  a  month,  in  3  yrs.  6  mos. 

160.  Find  the  interest  on  $1825  from  Jan.  1  to  June  25, 

at  5£%,  counting  the  exact  number  of  days,  and 
allowing  365  dys.  for  a  year. 

BANK  DISCOUNT 
Find  the  proceeds  of  the  following  notes : 

161.  $300.  SPRINGFIELD,  111.,  Aug.  12.  1884. 
Sixty  days  after  date  I  promise  to  pay  Nicholas  Welsh, 

or  order,  $300,  value  received. 

Discounted  at  6%,  Sept.  1.  JOHN  BRYCE. 

162.  $700.  BOSTON,  Nov.  13,  1880. 
Ninety  days  after  date  I  promise  to  pay  to  the  order  of 

David  Morrison  seven  hundred  dollars,  value  received. 
Discounted  at  7%,  Jan.  1,  1881.  GEORGE  BROWN. 

163.  $217.40.  NEW  YOEK,  July  30,  1884. 
Ninety  days  after  date  I  promise  to  pay  to  the  order  of 

Seth  Jay  two  hundred  seventeen  and  y4^  dollars,  value 
received. 

Discounted  at  6%,  Aug.  10,  1884.  JAMES  BENT, 


MISCELLANEOUS   PROBLEMS.  309 

164.  $500.  CHICAGO,  July  9,  1883. 
Ninety  days  from  date,  for  value  received,  I  promise  to 

pay  to  the  order  of  John  Hogan  five  hundred  dollars,  with 
interest  at  9%. 

Discount  at  6%,  July  9,  1883.  JOHN  FOSTER. 

165.  $5897.50.  TROY,  June  24,  1881. 
Four  months  from  date,  for  value  received,  I  promise  to 

pay  to  the  order  of  Aaron  Reed  five  thousand  eight  hun- 
dred ninety-seven  and  y5^  dollars,  with  interest  at  6%. 
Discounted  at  5%,  Aug.  15.  JAMES  CAREY. 

Find  the  face  of  a  note  which : 

166.  Discounted  at  6%  for  90  dys.  yields  $344.57. 

167.  Discounted  at  9%  for  46  dys.  yields  $493.87. 

168.  Discounted  for  6%  for  3  mos.  yields  $984.50. 


PARTIAL  PAYMENTS. 

169.  A  note  for  $680,  dated  June  15,  1884,  payable  on 

demand,  with  interest  at  6%,  hears  the  following 
endorsement:  May  15,  1885,  $425.  What  is  due 
June  15,  1885? 

170.  On  a  note  of  $1400,  dated  March  1,  1880,  there  was 

received  Oct.  19,  1880,  $700;  Jan.  1,  1881,  $400. 
What  is  due  March  1,  1881,  reckoning  interest  at 
6%? 

171.  A  note  of  $900,'  dated   Jan.   1,  1884,  and  bearing 

interest  at  5  °f0 ,  has  the  following  endorsements : 
May  13,  $240;  Aug.  19,  $300;  Oct.  25,  $180. 
Kequired  the  balance  due  Jan.  1,  1885. 


310  MISCELLANEOUS   PROBLEMS. 

172.  A  note  of  $1800,  dated  Jan.  1,  1880,  and  bearing 

interest  at  5%,  has  the  following  endorsement: 
June  1,  1881,  $400.  Find  the  balance  due  June 
1,  1884. 

173.  A  note  of  $600,  dated  Aug.  13,  1881,  and  bearing 

interest  at  6%,  has  the  following  endorsements: 
Jan.  1,  1882,  $200;  April  1,  1882,  $110.  Find 
the  balance  due  Aug.  13,  1883. 

174.  A  note  of  $1150,  dated  June  30,  1878,  and  bearing 

interest  at  6%,  has  the  following  endorsements: 
Jan.  30,  1879,  $15;  April  30,  1880,  $570;  July 
30,  1881,  $420.  Find  the  balance  due  Dec.  30, 
1882. 

COMPOUND  INTEREST. 
Find  the  compound  interest  of: 

175.  $300,  at  6%,  for  3  yrs.  4  mos.  18  dys. 

176.  $350,  at  6%,  for  3  yrs.  5  mos.  24  dys. 

177.  $840,  at  8%,  from  June  13,  1880,  to  Aug.  1,  1881, 

interest  being  payable  quarterly. 

178.  $400,  at  4|%,  from  Jan.  1,  1881,  to  Feb.  13,  1884. 

179.  $1100,  at  6%,  for  2  yrs.  7  mos.  6  dys.,  interest  being 

payable  semi-annually. 

180.  $1000,  at  8%,  for  2  yrs.  3  mos.  18  dys.,  interest  pay- 

able quarterly. 

STOCKS. 

181.  Find  the  cost  of  $2400  stock,  at  97f 

182.  Find  the  cost  of  $2785  stock,  at  105f 

183.  Find  the  cost  of  $5680  stock,  at  103 J. 


MISCELLANEOUS   PROBLEMS.  31  1 

184.  How  much  stock,  at  85f  ,  including  brokerage,  can 

be  bought  for  $23  76.84? 

185.  How  much    6%   stock  will   produce    an    income    of 

$840? 

186.  Find  the  price  of  stock,  when  $4647.50  will  pay  for 

$5200  worth  of  stock. 

187.  How  many  hundred-dollar  shares  of  1%  stock  will 

yield  a  yearly  income  of  $686? 

188.  A  gentleman    gave   his   daughter  $25,700  of  ±\% 

bonds.     What  yearly  income  from  them  will  she 
receive  ? 

189.  What  amount  of  8%  stock  will  yield  a  yearly  income 

of  $8000? 

190.  What  is  the  rate  of  dividend  when  the  sum  of  $300 

is  received  from  $  7500  stock  ? 

191.  Find  the  rate  of  dividend  when  the  sum  of  $1603.80 

is  received  from  $35,640. 

192.  How  much  stock,  at  121|,  can  be  bought  for  $6318? 

193.  How  much  stock,  at  97|,  can  be  bought  for  $1755? 

194.  Find  the  sum  paid  for  $5600  stock,  at    112|,  and 

brokerage  \. 

195.  What  income  will  be  obtained  from  $5125,  invested 

in  6%  stock,  at  102£? 

196.  Find  the  income  from  $8190,  invested  in  5%   stock, 

at  91. 

197.  Find  the  income  on  $1935,  invested  in  8%  stock,  at 


198.    Find  the  income  from  $6750  invested  in  4£  stock,  at 
75. 


312  MISCELLANEOUS   PROBLEMS. 

199.  If  $7656  be  invested  in  stock,  at  63$,  and  the  stock 

pays  a  dividend  of  3^%,  how  much  will    be  re- 
ceived on  the  money  invested  ? 

200.  If  $7000  be  invested  in  stock,  at  87£,  and  the  stock 

pays  a  dividend  of   7£%,  how  much  will  be  re- 
ceived ? 

201.  If  9%  stock  is  bought  at  150,  what  rate  of  interest 

will  be  received  on  the  investment? 

202.  What  rate  of  interest  will  be  received  on  5%  stock 

at  75? 

203.  What  rate  of  interest  will  be  received  on  4%  stock, 

at62|? 

204.  How  much  money  must  be  invested  in  5%   stock,  at 

80,  to  produce  $400  income? 

205.  How  much  money  must  be  invested  in  4%   stock,  at 

90,  to  produce  $320  income? 

206.  How  much  money  must  be  invested  in  6%  stock,  at 

75,  to  produce  $200  income? 

207.  A  man  received  $240  from   his    6%    dividend,  on 

stock  bought  at  105.     How  much  money  did  he 
have  invested  in  the  stock  ? 

208.  What  should  be  paid  for  a  4%  stock,  that  5%  inter- 

est may  be  realized  on  the  investment  ? 

209.  What  should  be  paid  for  a  6%  stock,  that  8%  inter- 

est may  be  realized  on  the  investment  ? 

210.  If  4%  stock,  which  produces  an  income  of  $180,  is 

sold  at  90,  what  sum  will  be  realized  from  the  sale  ? 

211.  What  increase  of  income  will  there  be,  if  $3600  of 

4%  stock  is  sold  at  90,  and  the  proceeds  invested 
in  7%  stock,  at  108? 


MISCELLANEOUS    PROBLEMS.  313 

212.  Find  the  increase  of  income,  if  $3900  of  3%   stock 

is  sold  at  88,  and  the  proceeds  invested  in  4-|% 
stock,  at  par. 

213.  Find  the  increase  of  income,  if  $6000  of  8%   stock 

is  sold  at  120,  and  the  proceeds  invested  in  6-^-% 
stock,  at  90. 

SIMPLE  PROPORTION. 

214.  If  16  bbls.  of  apples  cost  $28,  what  will  129  bbls. 

cost? 

215.  If  15  workmen  can  do  a  piece  of  work  in  25  dys.,  in 

how  many  days  can  25  men  do  the  same  work  ? 

216.  If  8  horses  eat  a  certain  quantity  of  hay  in  2  mos., 

how  long  will  the  same  quantity  last  12  horses? 

217.  A   meadow  can   be  mowed  by  40  men  in   10   dys. 

How  many  days  will  it  take  30  men  to  mow  it? 

218.  If  30  men  can  build  a  wall  in  18  dys.,  how  many 

men  will  be  required  to  build  it  in  12  dys.  ? 

219.  A  bankrupt  owes  $3000,  and  his  assets  amount  to 

$850.     How  much  on  a  dollar  will  his  creditors 
receive  ? 

220.  What  does  a  bankrupt  pay  on  a  dollar,  if  his  credit- 

ors receive  $376.275  on  $2076? 

221.  A  bankrupt's  effects  amounted  to  $2675.40,  and  his 

debts    to    $3057.60.       What    did    his    creditors 
receive  on  a  dollar? 

222.  If  4  men    reap  5  A.  159  sq.  rds.  in  1  week,  how 

many  men  at  the  same  rate  will  reap  35  A.  154 
sq.  rds.  ? 


314  MISCELLANEOUS    PROBLEMS. 


223.  A  wall  whose  height  is  9.1875  ft.  casts  a  shadow  of 

10.5  ft.  Find  the  length  of  the  shadow  of  a 
steeple  93.8  ft.  high. 

224.  A  cistern  can  be  filled  in  54  min.  by  a  pipe  running 

3J-  gals,  a  minute.  In  how  many  minutes  can  it 
be  filled  by  another  pipe,  running  4^-  gals,  a  minute? 

225.  A  watch  set  on  Saturday,  at  half-past  eight  in  the 

evening,  loses  1£  min.  in  30  hrs.  What  time  does 
it  show  the  next  Thursday,  at  4  o'clock  in  the 
afternoon  ? 

226.  When  do  the  hour  and   minute-hands   of  a  watch 

coincide  between  5  and  6  o'clock  ? 

NOTE.  Since  the  hour-hand  moves  through  5  minute-spaces  while 
the  minute-hand  traverses  60,  the  minute-hand  moves  12  times  as 
fast  as  the  hour-hand.  The  minute-hand,  therefore,  in  moving 
through  12  minute-spaces,  traverses  11  minute-spaces  more  than  the 
hour-hand. 

When  the  hour-hand  is  at  V.,  the  minute-hand,  being  at  XII.,  is 
25  minute-spaces  behind  it.  The  question,  therefore,  is,  if  the 
minute-hand,  to  gain  11  spaces,  must  move  through  12  spaces,  how 
many  spaces  must  it  move  through  to  gain  25  spaces  ? 

11  :  12  : :  25  :  ? 

227.  When  do  the  hour  and   minute-hands    of  a  watch 

coincide  between  8  and  9  o'clock  ? 

228.  When  do  the    hour  and    minute-hands    of  a  watch 

coincide  between  3  and  4  o'clock  ? 

229.  When  do  the    hour  and    minute-hands    of  a  watch 

coincide  between  10  and  11  o'clock? 

The  true  weight  of  a  body  weighed  successively  in  the 
scales  of  a  false  balance  is  the  square  root  of  the  product  of 
the  apparent  weights. 


MISCELLANEOUS   PROBLEMS  315 

230.  A  body  appears  to  weigh  5^g-  Ibs.  in  one  scale,  and 

5-$-  Ibs.  in  the  other  scale,  of  a  false  balance.  Find 
its  true  weight. 

The  times  in  which  bodies  fall  are  proportional  to  the 
square  roots  of  the  distances  traversed.  Since  a  body  falls 
16.1  ft.  the  first  second,  to  find  the  time  a  body  is  falling, 
divide  the  distance  by  16.1,  and  extract  the  square  root  of 
the  quotient. 

231.  In  how  many  seconds  will  a  stone  fall  to  the  bottom 

of  a  coal-pit  420  ft.  deep? 

COMPOUND  PROPORTION. 

232.  If  60  bu.  of  corn  feed  6  horses  for  50  dys.,  in  how 

many  days  will  15  horses  consume  75  bu.  ? 

233.  If  20  cwt.  are  carried  50  miles  for  $5,  how  much  will 

be  the  cost  of  carrying  40  cwt.  40  miles? 

234.  If  20  men  can  perform  a  piece  of  work  in  12  dys., 

required  the  number  of  men  who  can  perform  an- 
other piece  of  work  three  times  as  great  in  -J-  of 
the  time. 

235.  If  12  horses,  in  5  dys.,  draw  44  loads  of  stone,  how 

many  horses  will  draw  132  loads  the  same  dis- 
tance in  18  dys.  ? 

236.  If  a  footman  travels  130  mi.  in  3  dys.,  of  14  hrs. 

each,  in  how  many  days,  of  7  hrs.  each,  will  he 
travel  390  mi.  ? 

237.  If  50  men  dig  a  cellar  in  7  dys.,  working  11  hrs.  a 

day,  how  many  days  will  24  men  require,  work- 
ing 8  hrs.  a  day  ? 


316  MISCELLANEOUS   PROBLEMS. 

238.  A  garrison  of  1500  men  has  provisions  for  12  wks., 

at  the  rate  of  20  oz.  per  day  to  each  man.  How 
many  men  will  the  same  provisions  maintain  for 
20  wks.,  allowing  each  man  only  8  oz.  per  day? 

239.  If  12  candles,  of  which  8  weigh  a  pound,  serve  4 

winter  evenings,  from  five  to  eleven,  how  many  can- 
dles, of  which  6  weigh  a  pound,  will  serve  3  spring 
evenings,  from  seven  to  eleven  ? 

240.  A  contractor,  having  engaged  to  lay  10  mi.  of  rail- 

way in  150  dys.,  finds  that  90  men  have  finished 
3  mi.  in  80  dys.  How  many  more  men  must  he 
engage  to  finish  the  work  in  the  given  time? 

241.  If  200  men  in  12   dys.,  of  8  hrs.  each,  can   dig  a 

trench  160  yds.  long,  6  yds.  wide,  and  4  yds.  deep, 
in  how  many  days,  of  10  hrs.  each,  will  90  men 
dig  a  trench  450  yds.  long,  4  yds.  wide,  and  3  yds. 
deep? 

242.  If  120  men  make  an  embankment  f  of  a  mile  long, 

30  yds.  wide,  and  7  yds.  high,  in  42  dys.,  how 
many  men  will  it  take  to  make  an  embankment 
1000  yds.  long,  36  yds.  wide,  and  22  ft.  high,  in 
30  dys.  ? 

POWERS  AND  ROOTS. 
Find  the  square  root  of: 

243.  30976.  247.    2052.09. 

244.  106929.        248.  4795.25731. 

245.  622521.        249.  24674.1264. 

246.  1234321.       250. 


MISCELLANEOUS   PROBLEMS.  317 

Find  the  cube  root  of : 

251.  373248.  256.  52734.375. 

252.  54872.  257.  7834.87438. 

253.  389017.  258.  0.053157376. 

254.  1092727.  259.  f 

255.  84604519.  260.  7£. 

MENSURATION. 

261.  What  is  the  total  surface   of  a  cube,  the   edge   of 

which  measures  4i  in.? 

262.  How  many  planks,  each  15  ft.  long,  and  10  in.  wide, 

will  be  required  for  the  flooring  of  a  room  30  ft. 
in  length  and  22i  ft.  in  width? 

263.  A  square  court,  whose  side  is  42  yds.,  is  paved  with 

28,224  square  tiles.  Find  the  dimensions  of  each 
tile. 

264.  Find  the  area  of  a  triangle  whose  base  is  9  ft.  8  in., 

and  whose  altitude  is  5  ft.  3  in. 

265.  How  many  yards  of  carpeting,  1  yd.  wide,  will  be 

required  for  a  room  27  ft.  long,  and  21  ft.  3  in. 
wide,  if  the  strips  run  across  the  room  ? 

266.  How  many  yards  in  the  side  of  a  square  field  con- 

taining 3  A.  44  sq.  rds.  25  sq.  yds.  ? 

267.  The  weight  of  a  cubic  inch  of  water  is  253.17  grs. ; 

that  of  a  cubic  inch  of  air,  0.31  grs.  Find  to  three 
places  of  decimals  the  number  of  cubic  inches  of 
water  equal  in  weight  to  1  cu.  ft.  of  air. 

268.  How  many  cubic  feet  in  a  piece  of  timber  18  ft.  long, 

15  in.  wide,  and  10  in.  thick? 


318  MISCELLANEOUS   PROBLEMS. 

269.  The  sides  of  a  triangular  garden  are  52.64,  and  72 

yds.  respectively.     Find  the  area. 

270.  Find  the  circumference  and  area  of  a  circle  whose 

radius  is  2  ft.  4  in. 

271.  The  side  of  an  equilateral  triangle  is  12  ft.     Find  its 

area. 

272.  How  many  bricks,  each  9  in.  long,  4}  in.  wide,  and 

3  in.  thick,  will  be  required  for  a  wall  175  yds. 
long,  12  ft.  high,  and  1  ft.  10i  in.  thick? 

273.  The  diameter  of  a  circular  shaft  in  a  railway  tunnel 

is  5  ft.  and  its  depth  30  fathoms.  How  many 
cubic  feet  of  earth  were  dug  out  in  making  it? 

274.  Find  the  cost  of  plastering  a  room  25  ft.  6  in.  long, 

17  ft.  3  in.  wide,  and  10  ft.  8  in.  high.  The  price 
for  the  walls  is  21  cts.,  and  for  the  ceiling  32  cts. 
a  square  yard,  and  no  allowance  is  made  for  doors 
and  windows. 

275.  A  line  62  ft.  long  reaches  from  the  top  of  a  house  48 

ft.  high,  to  the  bottom  of  a  house  on  the  opposite 
side  of  the  street.  Find  the  width  of  the  street. 

276.  Find  the  capacity  in  bushels  of  a  round  basket  20  in. 

in  diameter  and  28  in.  deep. 

277.  The  diameter  of  a  spherical  balloon  is  25  ft.     How 

many  square  yards  of  silk  were  required  to  make 
it,  and  how  many  cubic  feet  of  gas  will  be  required 
to  fill  it? 

278.  Find  the  weight  of  an  ivory  ball  2  in.  in  diameter, 

the  weight  of  ivory  being  1825  oz.  a  cubic  foot. 

279v    Hew  many  cubic  feet  in  a  stone  roller  6  ft.  6  in.  long 
and  5  ft.  4  in.  in  circumference  ? 


CHAPTER  XVI. 

METRIC    MEASURES. 

320.  The  Metric  System  is  a  system  of  weights  and  meas- 
ures expressed  in  the  decimal  scale. 

321.  The  standard  meter,  as  defined  by  law,  is  the  length 
of  a  bar  of  very  hard  metal  carefully  preserved  at  Paris, 
accurate  copies  of  which  are  furnished  to  the  governments 
of  all  civilized  nations. 

322.  The  principal  units  of  the  metric  system  are  : 

The  meter  (m)  for  lengths  ; 
The  square  meter  (qm)  for  surfaces ; 
The  cubic  meter  (cbm)  for  large  volumes ; 
The  liter  (*)  (lee'-ter)  for  smaller  volumes ; 
The  gram  (g)  for  weights. 

323.  All  these  units  are   divided  and  multiplied  deci- 
mally, and  the  size  of  the  measures  thus  produced  is  shown 
by  one  of  seven  prefixes ;  namely,  deka,  meaning  10  ;  Jiekto, 
meaning  100  ;  kilo,  meaning  1000 ;  myria,  meaning  10,000  ; 
and  deci,  meaning  0.1 ;  centi,   meaning  0.01 ;  milli,  mean- 
ing 0.001. 

324.  But,  as  in  United  States  money  we  seldom  speak 
of  anything  else  than  dollars  and  cents,  so  in  other  meas- 
ures it  is  only  those  printed  in  black  letter  in  this  chapter 
that  are  in  common  use. 

NOTE.  A  meter  is  a  trifle  more  than  39.37  inches,  and  all  the  units 
of  the  system  are  derived  from  the  meter.  All  the  compound  names 
are  accented  on  the  first  syllable ;  thus,  millimeter.  The  teacher 
should  be  supplied  with  a  meter  stick,  a  liter,  and  a  cubic  centimeter. 


320  METRIC   MEASURES. 

UNITS  OF  LENGTH. 

325.  A  millimeter  (mm)  =  0.001  of  a  meter. 
A  centimeter  (cm;  ==  0.01 

A  decimeter  =0.1         "        " 
A  meter  (m).  Principal  Unit, 

A  dekameter  =         10  meters. 

A  hektometer  =       100      " 

A  kilometer  (km)  =    1,000      " 

A  myriameter  =  10,000      " 

326.  A  length  given  in  any  one  of  these  meas- 
ures may  be  expressed  in  terms  of  another  measure 
by  simply  moving  the  decimal  point  to  the  right 
or  left. 

Thus,  17,856,342mm  may  be  written  as  kilo-me- 
ters by  observing  that  milli- meters  are  changed  to 
meters  by  moving  the  point  three  places  to  the 
left ;  and  these  meters  into  kilo-meters  by  carrying  5 
it  three  places  further,  making,  in  all,  six  places,   f 
Therefore,  17,856,342°"*  =  17.856342km. 

Again,  4.876326km  may  be  written  as  centi-  « 
meters,  by  observing  that  kilo-meters  are  changed  I 
to  meters  by  moving  the  point  three  places  to  the  I 
right,  and  meters  to  centi-meters  by  moving  it  two  « 
places  further,  making,  in  all,  five  places.  There- 
fore, 4.876326km  =  487,632.6cm. 

327.  The  rule,  therefore,  for  this  conversion  is  : 
First  change  the  point  so  as  to  convert  the  given 

measures  into  terms  of  the  principal  unit ;  then 
change  the  point  so  as  to  convert  the  principal 
into  the  required  units. 

328.  Remember    that,   before    adding    or    sub- 
tracting,  the   quantities   must   be   written  in  the 
same  units  of  measure. 


r. 


if 


r 


METRIC   MEASURES.  321 

Ex.  165.     (Oral.) 

1.  How  many  meters  in  a  dekameter  ?  hectometer?  kilo- 

meter? How  many  dekameters  in  a  hectometer? 
kilometer  ? 

2.  What  part  of  a   meter  is  a  decimeter?  centimeter? 

millimeter  ?  What  part  of  a  centimeter  is  a  milli- 
meter ? 

3.  Read32.3m;   12.6cm;   15.4km ;  59.8mm. 

4.  Express  3256m  as  kilometers  ;  as  centimeters. 

5.  Express  5368mm  as  centimeters  ;  as  meters. 

6.  Express  12.4km  as  meters;  as  centimeters. 

Ex.  166. 

Find  the  value  of  each  of  the  following  expressions  in 
meters  : 

1.  0.435m  +  852cm  +  4263mra  +  0.1595km. 

2.  0.927kra  —  6495cm  ;  4.37cra  —  42.87ram. 

3.  8  X  0.0457km ;  3.04  X  60.93cm  ;  5.43  x  67.2ram. 

4.  38,019mm^  0.097;  0.41km  -*-  25.625. 

5.  A  book  is  2.1cm  thick ;  if  the  average  thickness  of  the 

leaves  is  0.05mm,  find  the  number  of  pages  in  the  book. 

6.  The  expense  of  building  a  certain  railroad  is  $25,000 

on  the  average  per  kilometer.  What  is  the  whole 
cost  of  the  road,  if  its  length  is  72km  and  53m. 

7.  The  wheels  of  a  locomotive  that  makes  45km  an  hour 

are  7.5m  in  circumference.  How  many  revolutions 
will  they  make  a  minute  ? 

8.  A  locomotive  runs  1284m  in  1}  min.    How  many  kilo- 

meters will  it  go  in  1  hr.  35  min.  15  sec.  ? 

9.  The  top  of  a  monument  is  143.9m,  and  the  base  67.19m 

above  the  level  of  the  sea;  the  steps  which  lead 
from  the  base  to  the  top  of  the  monument  are  each 
19cm  high.  How  many  steps  are  there  ? 


322  METRIC   MEASURES. 


I  I  I  I  I  I  i~| 


MEASURES  OF  SURFACE. 

329.  The  principal  unit  of  surface  is  a  square  meter  (qm). 

330.  In  square  measure  the  multiplication  and  division 
of  units   is  by   hundreds  and  hundredths, 

instead  of  by  tens  and  tenths'  Suppose  the 
square  in  the  margin  to  represent  a  square 
meter,  It  is  divided  into  ten  equal  horizon- 
tal bands,  and  each  band  is  one-tenth  of  the 
square  meter.  Each  band  can  be  divided, 
as  the  upper  one  is,  into  ten  little  squares  measuring 
one-tenth  of  a  meter  on  a  side.  Each  of  these  squares  will 
be  0.1  of  the  band,  or  0.01  of  the  whole  square.  The  square 
meter,  therefore,  contains  10  X  10  or  100  square  decimeters. 

If  the  square  meter  were  divided  into  100  equal  hori- 
zontal bands,  each  band  would  be  0.01  of  the  square ;  and 
if  each  of  the  100  bands  were  divided  into  100  squares,  that 
is,  into  100  square  centimeters,  the  whole  square  would 
contain  100  X  100  or  10,000  square  centimeters.  A  square 
meter,  therefore,  contains  10,000  square  centimeters, 

In  like  manner,  a  square  meter  contains  1,000,000  square 
millimeters, 

331.  UNITS  OF  SURFACE. 

A  square  millimeter  (qmm)  —  0.000001  of  a  square  meter. 

A  square  centimeter  (qcm)     —  0.0001 

A  square  decimeter  =  0.01 

A  square  meter  (qm)  Principal  Unit, 

A  square  dekameter  100  square  meters. 

A  square  hektometer         =       10,000       "  " 

A  square  kilometer  (qkm)       =  1,000,000       " 

332.  It  will  be  observed  that  while  centimeters  are  in 
the   second,    a*nd   millimeters   in  the  third  decimal  place 
from  meters,  square  centimeters  are  in  the  fourth  and  square 
millimeters  in  the  sixth  decimal  place  from  square  meters, 


METRIC   MEASURES.  323 

LAND  MEASURE. 

333.  In  measuring  land  the  square  meter  is  called  a 
centar  (ca),  the  square  dekameter  is  called  an  ar  (a),  and  the 
square  hektometer  a  hektar  (ha). 

Ex.  167.     (Oral.) 

1.  How  many   square  meters  in    a   square    dekameter? 

square  hectometer  ?  square  kilometer  ? 

2.  How  many  centars  in  an  ar  ?  in  a  hektar  ? 

3.  What  part  of  a  hektar  is  an  ar  ?  a  centar  ? 

4.  What  part  of  a  square  meter  is  a  square  decimeter? 

square  millimeter  ? 

5.  Read56.4qra;  2.05qkm;  531.6qcm. 

6.  Read53a;  36.03ha;  56ca;  56qm. 

Ex  168. 

1.  Convert  1,854, 276qm  into  hektars;    into  square   kilo- 

meters. 

2.  How  many  hektars  in  2. 7856qkm? 

3.  Write  1.7431qm  as  square  centimeters;  as  square  milli- 

meters. 

4.  How  many  square  kilometers  in  17,467. 5ha? 

5.  How  many  square  meters  in  1.3614qkm? 

6.  How  many  square  meters  in  2.25ha? 

7.  How  many  square  centimeters  in  0.0137  of  a  square 

meter  ? 

8.  Write  3.571qcm  as  square  millimeters. 

9.  A  man  bought  3ba  of  land  at  $200  per  hektar,  and  sold 

it  for  $2.50  per  ar.     How  much  did  he  gain  ? 
10.    If  6ha  are  divided  into  64  equal  lots,  how  many  square 
s  meters  will  there  be  in  each  lot? 


324 


METRIC   MEASURES. 


MEASURES  OF  VOLUME. 

334.  The  principal  unit  of  capacity  or  volume  is  a  cubic 
meter  (cbm). 

335.  The  cubic  meter   can   be  divided  into   10   layers, 
each  a  meter  square  and  a  deci- 
meter thick.     Each  layer  will, 

therefore,  be  0.1  of  a  cubic  meter. 

Again,  each  layer  can  be 
divided  into  10  equal  parts. 
Each  part  will,  therefore,  be 
0.1  of  the  layer,  or  0.01  of  the 
meter,  and  will  be  a  decimeter 
square  and  a  meter  long. 

Also,  each  one  of  these  parts 
can  be  divided  into  10  equal  parts,  each  of  which  will  be  a 
cubic  decimeter,  and  will  be  0.1  of  0.01,  that  is,  0.001  of 
the  cubic  meter. 

The  cubic  meter,  therefore,  contains  1000  cubic  decimeters, 

In  like  manner,  each  cubic  decimeter  can  be  divided  into 
1000  cubic  centimeters,  and  each  cubic  centimeter  into  1000 
cubic  millimeters. 

336.  UNITS  OF  VOLUME. 

A  cubic  millimeter  (cmm)  =  0.000000001  of  a  cubic  meter. 
A  cubic  centimeter  (oom)     —  0.000001  "      " 

A  cubic  decimeter  =  0.001  "      " 

A  cubic  meter  (cbm)  Principal  Unit, 

337.  It  will  be  seen  that  cubic  centimeters  are  in  the  sixth 
and  cubic  millimeters  are  in  the  ninth  decimal  place  from 
cubic  meters, 

338.  In  measuring  wood,  the  cubic  meter  is  called  a  ster 
(st) ;  and  in  measuring  liquids,  grain,  etc.,  the  cubic  deci- 
meter is  always  called  a  liter, 

When  the  liter  is  the  unit,  the  numeral  prefixes  have  the 
bame  value  as  in  linear  measure. 


METRIC   MEASURES.  325 


Ex.  169.     (Oral.) 

1.  In  a  cubic  meter,  how  many  cubic  decimeters?  cubic 

centimeters  ? 

2.  What  part  of  a  cubic  meter  is  a  cubic  decimeter? 

3.  What  part  of  a  cubic  meter  is  a  cubic  centimeter  ? 

4.  What  part  of  a  cubic  decimeter  is  a  cubic  centimeter? 

5.  How  many  liters  in  a  hektoliter  ? 

6.  How  many  liters  in  a  cubic  meter? 

7.  How  many  cubic  centimeters  in  a  liter? 

8.  Express  7685. 251  as  cubic  meters ;  as  hektoliters. 

9.  If  the  water  in  the  liter  represented  in  the  margin 

stands  6cm  high,  how  many 
cubic  centimeters  of  water 
are  there  in  the  measure? 
How  many  will  be  required 
to  fill  it?  If  the  faucet  be 
turned,  and  the  water  allowed 
to  run  out  until  the  measure 
Liter  =  cubic  Decimeter.  is  only  a  quarter  full,  how 

many  cubic  centimeters  will  run  out?     How  many 

will  still  remain? 

Ex.  17O. 

1.  How  many  cubic  meters  in  a  rectangular  box  125cm 

long,  IIS0™  wide,  and  80cm  deep?  how  many  liters? 

2.  How  many  cubic  meters  of  earth  must  be  removed  to 

dig  a  ditch  90ra  long,  85cm  wide,  and  50cm  deep? 

3.  How  deep  must  a  cistern  be  to  hold  60001,  if  the  bot- 

tom is  a  square  measuring  2.25™  on  a  side  ? 

4.  How  many  hektoliters  in  a  bin  4m  long,  2m  wide,  and 

lmhigh? 

6.    How  high  must  a  box  be  to  hold  301,  if  it  is  50cm  long 
and  20cm  wide? 


326  METRIC   MEASURES. 

UNITS  OF  WEIGHT. 

339.  The  units  of  weight  are  the  weights  of  units  of  pure 
water  taken  at  its  greatest  density, 

that  is,  a   little  above    the    freezing 
point. 

The   principal   unit    is    the   gram, 
which  is  the  weight  of  a  cubic  centi- 

Cubic  Centimeter.  Gram  Weight 

meter  of  water. 

The  numeral  prefixes  have  the  same  value  as  in  linear 
measure. 

340.  A  cubic  centimeter  of  water  weighs  a  gram. 
A  liter  of  water  weighs  a  kilogram, 

A  cubic  meter  of  water  weighs  a  ton  (1000kg). 

Ex.  171.      (Oral.) 

1 .  How  many  grams  in  a  kilogram  ?  dekagram  ?  hektogram  ? 

2.  What  part  of  a  gram  is  a  centigram?  decigram?  milligram? 

3.  In  a  ton,  how  many  kilograms?  grams? 

4.  A  hektoliter  of  water  weighs  how  many  kilograms? 

what  part  of  a  ton  ? 

5.  What  is  the  weight  in  grams  of  5ccm  of  water? 

6.  Change  12,260rag  into  grams. 

7.  Give  the  weight  in  kilograms  of  0.275cbm  of  water. 

8.  Change  0.546kg  to  grams  ;  to  milligrams. 

9.  Change  0.563  of  a  ton  to  kilograms. 

Ex.  172. 

1.  What  is  the  weight  of  water  required  to  fill  a  vat  98cm 

long,  71cm  wide,  and  38cm  deep  ? 

2.  A  mass  of  21. 7g  is  divided  into  70  pills      What  is  the 

weight  of  each  pill  ? 

3.  At  2  cts.  a  kilogram,  what  will  2.25  tons  of  hay  cost  ? 

4.  At  $6  a  ton  for  coal,  what  will  it  cost  to  heat  a  building 

30  days,  if  it  takes  400kg  of  coal  a  day  ? 


METRIC  MEASURES.  327 

SPECIFIC  GRAVITY. 

341.  The  specific  gravity  of  any  substance  is  the  number 
found  by  dividing  the  weight  of  the  substance  by  the  weight 
of  an  equal  bulk  of  water. 

342.  Therefore  the  specific  gravity  of  a  substance  is  the 
number  that  expresses  the  weight  of  a  cubic  centimeter  of  it 
in  grams ;  or  of  a  liter  in  kilograms ;  or  of  a  cubic  meter  in 
tons, 

343.  The  volume   of  a  body  is  found  by   dividing   its 
weight  by  its  specific  gravity. 

Ex.  173. 

1.  A  bar  of  iron  50cra  long,  4cm  wide,  lcm  thick  has  a  spe- 

cific gravity  of  7.8.     Find  its  weight  in  kilograms. 

2.  A  piece  of  iron  weighing  117kg  is  made  into  a  bar  6cm 

wide  and  2cm  thick.  What  is  its  length,  if  the  spe- 
cific gravity  of  the  iron  is  7.8  ? 

3.  A  cubical  vessel,  40cm  on  an  edge,  is  full  of  water.     If 

14l  are  drawn  off,  and  replaced  by  a  liquid  of  which 
the  specific  gravity  is  %  of  that  of  water,  what  is  the 
weight  of  the  mixture  ? 

4.  What  will  be  the  weight  in  air,  in  water,  and  in  olive 

oil,  of  which  the  specific  gravity  is  0.915,  of  a  cube 
of  iron  6cm  on  an  edge,  if  the  specific  gravity  of  the 
iron  is  7.8? 

NOTE.   A  body  weighed  in  a  liquid  weighs  less  than  in  air 
by  the  weight  of  the  volume  of  the  liquid  which  it  displaces. 

5.  What  is  the  specific  gravity  of  a  substance  of  which 

7.3ccm  weighs  31.5g? 

6.  A  block  of  granite  60cm  long,  50cm  wide,  15cm  thick 

weighs  130. 5kg.     Find  its  specific  gravity. 


328  METRIC  MEASURES. 


7.  What  is  the  weight  of  a  load  of  250  paving  stones, 

15cm  square  and  8cm  thick,  if  the  specific  gravity  of 
the  stone  is  2.90? 

8.  A  stick  of  timber  8m  long,  25cm  wide,  and  23cm  thick  is 

carried  by  4  men.  What  weight  does  each  carry,  if 
the  specific  gravity  of  the  timber  is  0.53  of  that  of 
water  ? 

9.  Allowing  that  water  in  freezing  is  increased  by  y1^  of 

its  volume,  find  the  volume  and  the  weight  of  a  block 
of  ice  SO0"1  long,  50cm  wide,  20cm  thick,  and  find  how 
much  water  it  will  give  on  melting. 

10.  A  hall  has  a  capacity  of  450cbm.    Compute  the  weight 

of  oxygen  in  the  air  that  fills  the  hall,  knowing  that 
air  contains  23%  of  its  weight  of  oxygen,  and  weighs 
y^  as  much  as  water. 

11.  A  bottle  empty  weighs  650g ;   full  of  oil,   it  weighs 

1075g.  What  part  of  a  liter  will  the  bottle  hold,  if 
the  specific  gravity  of  the  oil  is  0.905  ? 

12.  A  vessel  full  of  water  weighs  9.68kg,  and,  full  of  oil, 

of  which  the  specific  gravity  is  0.91,  weighs  9.266kg. 
Find  its  capacity  and  its  weight  when  empty. 

NOTE.  To  find  the  capacity,  divide  the  difference  of  the  two 
weights  by  the  difference  of  the  weights  of  a  liter  of  water 
and  a  liter  of  oil. 

13.  In  a  decimeter  cube  full  of  water  is  placed  a  chain  oi 

iron,  of  which  the  specific  gravity  is  7.8.  When  the 
chain  is  removed,  the  height  of  the  water  is  just  5cm. 
Find  the  volume  and  weight  of  the  chain. 

14.  If  a  body  weighs  3.71kg  in  air,  and  2.38kg  in  water, 

what  is  its  specific  gravity  ? 

15.  From  106. 25g  of  rhubarb  are  made  125  powders.    What 

is  the  volume  of  water  of  which  the  weight  is  equal 
to  the  weight  of  a  powder  ? 


METRIC   MEASURES.  329 

Ex.  174. 
MISCELLANEOUS  PEOBLEMS 

1.  A  well  is  18. 2m  deep  and  the  wheel   is   1.4m  round. 

How  many  turns  of  the  wheel  will  be  required  to 
raise  the  bucket? 

2.  Find  the  number  of  liters  in  a  vat  2m  by  75cm  by  50cm. 

3.  Into  how  many  pills  of  325mg  each  can  a  mass  of  23. 4g 

be  made  ? 

4.  How  many  liters  will  a  box  hold  which  is  75cra  long, 

15cm  wide,  and  12cradeep? 

5.  Add  3.473m,  SO8",  83mm,  4.5m,  and  16cm. 

6.  There  are  4  measuring  lines:    the   first  is   7.5m,  the 

second  is  3m  75cm,  the  third  is  4m  80cm,  and  the 
fourth  is  8m  6cm.  Express  in  meters  the  total  length 
of  the  four  lines. 

7.  On  the  same  railroad  are  four  stations,  between  which 

the  consecutive  distances  are  as  follows :  7km  249m, 
3km  200™,  and  5.007km.  Find  in  kilometers  the  dis- 
tance between  the  first  and  fourth  stations. 

8.  A  goldsmith  has  sold  jewels  of  the  following  weights 

respectively :  27g  9mg,  30g  70cg,  7g  4cg,  and  19*  34cg 
7mg.  Find  the  total  weight  in  grams. 

9.  From  17km  6m  take  243m  691mm. 

10.  From  a  farm  containing  340ha  7a  there  are  sold  119h8 

29.03a;  how  many  hektars  are  left? 

11.  A  liter  of  mercury  weighs  13kg  598g.     Find  the  weight 

in  kilograms  of  3.69  liters. 


330  METRIC   MEASURES. 


12.  If  16.941  of  olive  oil  weigh  15kg  500*,  find  the  weight 

of  one  liter. 

13.  Into  how  many  lots  of  3.75a  may  8ha  40a  be  divided? 

14.  If  122. 6g  of  chlorate  of  potassa  yield  48*  of  oxygen, 

what  weight  of  oxygen  may  be  obtained  from  lkg 
of  the  chlorate  ? 

15.  A  field  of  wheat  containing  8£ha  furni.rhed  600  sheaves 

per  hektar ;  2  sheaves  of  wheat  furnished  a  bundle 
of  straw  weighing  5kg.  What  will  the  whole  straw 
bring,  at  $15  a  ton? 

16.  A  farmer  shears  620  sheep  and  180  lambs ;  the  sheep 

give  on  an  average  4kg  of  wool  each,  and  the  lambs 
J  of  a  kilogram  each.  The  wool  is  sold  for  50  cts. 
a  kilogram.  How  much  money  does  the  farmer 
receive  ? 

17.  A  vessel  full  of  water  weighs  5.25kg;    the  weight  of 

the  vessel  when  empty  is  250g.  How  many  liters 
will  the  vessel  hold  ? 

18.  Wheat  weighs  80kg  a  hektoliter.     A  field  of  4.6ba  has 

produced  9.2  t.  of  wheat.  If  a  sheaf  of  the  wheat 
on  the  average  gives  41  of  grain,  find  the  average 
number  of  sheaves  produced  per  hektar. 

19.  A  piece  of  zinc  weighs  in  the  air  343g,  and  in  water 

293*  only.     What  is  its  volume  ? 

20.  A  piece  of  zinc  weighs  in  the  air  343g,  and  in  water 

293*.     What  is  its  specific  gravity  ? 

21.  A  jug  empty  weighs  1.02kg;    full  of  water  it  weighs 

3.8kg.     Find  the  capacity  of  the  jug  in  liters. 

22.  The  price  of  8  casks  of  olive  oil,  containing  each  9.05hl, 

is  $  1072.     What  will  be  the  price  of  201  ? 


ADVERTISEMENTS. 


WENTWORTH'S  ARITHMETICS, 

Adopted  for  exclusive  use  in  the  State  of  Washington,  and  in  countless 
cities*  towns,  and  schools. 


MASTERY:  their  motto. 

LEARN  TO  DO  BY  DOING:  their  method. 

PRACTICAL  ARITHMETICIANS:  the  result. 


WENTWORTH'S  PRIMARY  ARITHMETIC. 

By  G.  A.  WENTWORTH,  Professor  of  Mathematics  in  Phillips  Exe 
ter  Academy,  and  Miss  E.  M.  REED,  Principal  of  the  Training 
School,  Springfield,  Mass.  Profusely  illustrated.  Introduction 
price,  30  cents;  allowance  for  old  book  in  exchange,  10  cents, 

In  a  word,  this  book —  the  fruit  of  the  most  intelligent  and  pains- 
taking study,  long-continued  —  is  believed  to  represent  the  best 
known  methods  of  presenting  numbers  to  primarians,  and  to  pre- 
sent these  methods  in  the  most  available  form.  It  is  commended 
as  profoundly  philosophical  in  method,  simple  and  ingenious  in 
development,  rich  and  varied  in  matter,  attractive  in  style,  and  prac- 
tical in  effect. 

It  has  been  carefully  and  critically  examined  by  myself  and  my  teachers,  and  in  our 
estimation  it  stands  ahead  of  anything  else  of  the  kind  that  we  have  found.  —  PRINCIPAL 
CAMPBELL,  State  Normal  School*  Johnson,  Vt. 

WENTWORTH'S  GRAMMAR  SCHOOL  ARITHMETIC. 

Illustrated.  Introductory  price,  65  cents;  allowance,  20  cents. 
Answers  free  on  teachers'  orders. 

Intended  to  follow  the  Primary  Arithmetic  and  make  with  that  a 
two-book  series  for  common  schools.  It  is  designed  to  give  pupils 
of  the  grammar  school  age  an  intelligent  knowledge  of  the  subject 
and  a  moderate  power  of  independent  thought,  by  training  them  to 
solve  problems  by  neat  and  intelligent  methods  and  keeping  them 
free  from  set  rules  and  formulas.  It  is  characterized  by  accuracy, 
thoroughness,  good  sense,  school-room  tact,  and  practical  ingenuity. 

Eminently  practical,  well  graded,  and  well  arranged.  ...  I  consider  it  the  brightest, 
most  attractive,  most  scholarly  text-book  on  this  subject  that  has  been  issued  for  years. 
—  PRINCIPAL  SERVISS,  Amsterdamt  N.Y. 

In  a  word,  these  books  represent  the  Best  Methods,  made  feasible, 
With  the  Best  Problems, — ingenious,  varied,  practical,  and  abundant 


GINN  &  COMPANY,   Publishers, 

BOSTON,  NEW  YORK,  CHICAGO,  AND  LONDON., 


/T\usi<;al  publications. 

Introd. 
Caswell  &  Ryan  :  Time  and  Tune  Series.  Price. 

Book    I.     TheyEolian #0.65 

Book  II.     The  Barcarolle 94 

Coda Supplementary  Music  for  Public  Schools.    Send 

for  Catalogue. 

Eichberg Girls'  High  School  Music  Reader 1.25 

New  High  School  Music  Reader 94 

High  School  Music  Reader  (old  edition) 94 

Eichberg  &  Sharland  :  Fourth  Music  Reader  (Revised) 94 

Abridged  Fourth  Music  Reader  (Revised) 75 

Emerson,  Brown  &  Gay  :  The  Morning  Hour 50 

Leib Voices  of  Children 40 

Mason New  First  Reader 25 

New  Second  Reader 40 

New  Third  Reader 40 

Independent  Reader. .    70 

Abridged  Independent  Reader 60 

National  Music  Teacher 40 

Hymn  and  Tune  Book  for  Female  Voices 60 

Hymn  and  Tune  Book  for  Mixed  Voices 60 

Independent   and    Hymn    and   Tune    Book    for 

Mixed  Voices  (combined) 94 

New   First,  Second,  and  Third  Series  of  Music 

Charts each  9.00 

Mason  &  Veazie  :  New  Fourth  Music  Reader 90 

National  Music  Course.     See  Mason,  Mason  &  Veazie,  Eichberg, 
Eichberg  &  Sharland. 

Pease Singing-Book 70 

Tilden Common  School  Song  Reader 36 

Common  School  Chart 5.00 

Handbook  of  First-Year  Lessons 10 

Veazie Music  Primer 05 

Four-Part  Song  Reader 40 

Young Institute  Song  Collection 10 

Zuchtmann  &  Kirtland  :  Choral  Book 60 


GINN  &  COMPANY,  Publishers, 

BOSTON,  NEW  YORK,  &  CHICAGO. 


CLASSICS    FOR    CHILDREN. 


Choice  Literature;   Judicious  Notes;   Large  Type;  Firm  Binding; 
Low  Prices. 


Hans  Andersen's  Fairy  Tales. 

*  FIRST  SERIES  :  Supplementary  to  the  Third  Reader. 

*  SECOND  SERIES  :  Supplementary  to  the  Fourth  Reader. 
*-3£sop's  Fables,  with  selections  from  Krilof  and  La  Fontaine. 
*Kingsley's  Water-Babies  :  A  story  for  a  Land-Baby. 
*Ruskin's  King  of  the  Golden  River :  A  Legend  of  Stiria. 
*The  Swiss  Family  Robinson.     Abridged. 

Robinson  Crusoe.     Concluding  with  his  departure  from  the  island. 
*Kingsley's  Greek  Heroes. 

Lamb's  Tales  from  Shakespeare.    "  Meas.  for  Meas."  omitted. 

Martineau's  Peasant  and  Prince. 

Bunyan's  Pilgrim's  Progress. 

Scott's  Marmion ;   Lady  of  the  Lake ;  Lay  of  the  Last  Minstrel. 

Lamb's  Adventures  of  Ulysses. 

Tom  Brown  at  Rugby. 

Church's  Stories  of  the  Old  World. 

Scott's  Quentin  Durward.     Slightly  abridged. 

Irving's  Sketch  Book.     Six  Selections,  including  "  Rip  Van  Winkle." 

Irving's  Alhambra. 

Shakespeare's  Merchant  of  Venice. 

Scott's  Old  Mortality ;  Ivanhoe ;  Talisman ;  Rob  Roy ;  Guy  Man- 
nering ;  Tales  of  a  Grandfather.     Each  complete. 

Johnson's  Rasselas  :  Prince  of  Abyssinia. 

Gulliver's  Travels.     The  Voyages  to  Lilliput  and  Brobdingnag. 
*Plutarch's  Lives.     From  dough's  Translation. 

Irving-Fiske's  Washington  and  His  Country. 

Goldsmith's  Vicar  of  Wakefield. 
*Franklin :  His  Life  by  Himself. 

Selections  from  Ruskin. 

Heroic  Ballads. 
"Bale's  Arabian  Nights. 

Grote  and  Segur's  Two  Great  Retreats. 


Starred  books  are  illustrated. 


CINN   &   COMPANY,    Publishers, 

BOSTON,  NEW  YORK,  AND  CHICAGO. 


A   REVOLUTION    IN    SCHOOL    READING 


HAS  BEEN  WROUGHT  BY  THE  USE  OF  THE 

Classics  for  Children. 


The  books  in  this  carefully  edited  series  are  widely  used 
in  place  of  the  ordinary  Reading  Books  in  the  upper  grades 
of  the  Grammar  Schools  and  in  the  High  Schools.  They 
are  also  used  as  Supplementary  Readers  in  hundreds  of 
schools  throughout  the  country. 

DESIGN  — 

To  supply  material  for  practice  in  reading,  form  a  taste  for 
good  literature,  and  increase  the  mental  power  of  the  pupils 
by  providing  them  with  the  best  works  of  standard  authors, 
complete  as  far  as  possible,  and  judiciously  annotated. 

AUTHORSHIP  — 

Varied,  and  of  world-wide  reputation.  In  the  list  of  authors 
are  Shakespeare,  Ruskin,  Scott,  Irving,  Goldsmith,  Johnson, 
Franklin,  Andersen,  Kingsley,  De  Foe,  Swift,  Arnold,  and  Lamb. 

EDITORS  — 

Of  recognized  ability  and  discriminating  taste.  Among  them 
are  John  Fiske,  Edward  Everett  Hale,  Henry  N.  Hudson, 
Charlotte  M.  Yonge,  John  Tetlow,  Homer  B.  Sprague,  D.  H. 
Montgomery,  Edwin  Ginn,  W.  H.  Lambert,  Alfred  J.  Church, 
Dwight  Holbrook,  J.  H.  Stickney,  Margaret  A.  Allen,  and  Mary 
S.  Avery. 

INDORSED  BY  — 

Teachers,  Superintendents,  Librarians,  eminent  Literary 
Authorities,  and  the  Educational  Press 


OPEN  SESAME! 

About  One  Thousand  Pieces  of  the  Choicest  Prose  and  Verse. 

COMPILED  BY 
BLANCHE  WILDER  BELLAMY  AND  MAUD  WILDER  GOODWIN. 

VOL.  I.  for  children  from  four  to  ten  years  old. 
VOL.  II.  for  children  from  ten  to  fourteen  years  old. 
VOL.  III.  for  children  of  a  larger  growth. 

Illustrated,  and  handsomely  boiind  in  cloth.     Price  of  each  to 
teachers,  and  for  introduction,  75  cents. 


No  Eastern  romancer  ever  dreamed  of  such  a  treasure-house 
as  our  English  literature. 

With  this  "  Open  Sesame  "  in  his  possession  a  boy  or  girl 
has  only  to  enter  and  make  its  wealth  his  own. 

Every  piece  is  believed  to  be  worth  carrying  away  in  the 
memory. 

The  best  writings  of  our  classic  authors  are  here,  with  selec- 
tions from  recent  literature  and  not  a  few  translations. 

It  is  very  good  indeed.  We  think  it  the  best  of  all  the  collections.  —  E.  A. 
SHELDON,  Prin.  State  Normal  School,  Oswego,  N.Y. 

I  think  it  by  far  the  best  collection  of  memory  pieces  I  have  ever  seen.  — 
F.  B.  PALMER,  Prin.  State  Normal  School,  Fredonia,  N.Y. 

It  is  a  beauty,  and  of  all  similar  works  I  have  seen,  it  has  the  most  desira- 
ble selections.  —  W.  E.  BUCK,  Supt.  Public  Schools,  Manchester,  N.H. 

The  book  is  a  handsome  specimen  of  the  arts  of  typography  and  binding, 
while  the  selections  and  their  arrangement  speak  well  for  the  judgment  and 
taste  of  the  editors.  —  CHAS.  W.  COLE,  Supt.  Public  Schools,  Albany,  N.Y. 

It  [Volume  I.]  is  a  rare  and  rich  collection  of  poems  and  a  few  prose 
articles.  —  I N TER-OCEAN,  Chicago. 

The  whole  book  is  full  to  overflowing  of  the  best  things  to  be  found  in  the 
English  language,  and  is  a  thoroughly  happy  production  which  children, 
parents,  and  teachers  will  welcome  eagerly.  • —  EDUCATION,  Boston. 

It  is  not  often  that  a  collection  of  verse  so  thoroughly  representative  of  what 
is  best  in  literature,  and  so  inclusive  of  what  one  has  learned  to  love  and  to 
look  for  in  every  anthology,  comes  from  the  press.  —  CHRISTIAN  UNION, 
New  York. 

The  editors  have  brought  to  their  task  a  sufficiently  wide  and  sympathetic 
knowledge  of  English  and  American  verse,  and  have  also  wisely  considered  the 
real  needs  and  tastes  of  children. .  .  .  The  collection  is  at  once  of  a  high  char- 
acter and  of  a  practicable  sort.  —  SUNPAY  SCHOOL  TIMES.  Philadelphia. 


CINN    &   COMPANY,    Publishers, 

BOSTON,  NEW  YORK,  CHICAGO,  AND  LONDON. 


STICKNEY'S  READERS. 

Introductory  to  Classics  for  Children.  By  J.  H.  STICKNEY,  author  of  Tht 
Child's  Book  of  Language,  letters  and  Lessons  in  Language,  English 
Grammar,  etc.  Introduction  Prices:  First  Reader,  24  cents;  Second 
Reader,  32  cents;  Third  Reader,  40  cents;  Fourth  Reader,  50  cents; 
exchange  allowances  respectively  of  5  cents,  8  cents,  10  cents,  and  10 
cents.  Auxiliary  Books :  Stickney  &  Peahody's  First  Weeks  at  School, 
12  cents;  Stickney's  Classic  Primer,  20  cents. 

THESE  books  are,  first  of  all,  readers.  This  main  purpose  is 
not  sacrificed  in  order  to  get  in  all  sorts  of  "  features  "  to  entrap 
the  unwary. 

The  vitality  of  methods  and  selections  preserves  the  chil- 
dren's natural  vivacity  of  thought  and  expression. 

The  editor  aimed  at  positive  excellence,  and  not  simply  to 
make  a  series  so  characterless  that  no  one,  however  unreason- 
able or  ill-informed,  could  discover  a  feature  definite  enough 
to  find  fault  with. 

This  is  almost  the  only  series  that  contains  a  sufficient  quan- 
tity of  reading  matter,  and  there  is  no  padding. 

Good  reading  would  not  be  good  if  it  did  not  appeal  to  what 
is  good  in  us,  and  the  lessons  in  Stickney's  Readers,  without 
"  moralizing,"  carry  moral  influence  in  warp  and  woof. 

Give  the  children  a  chance  at  these  Readers.  They  are  the 
ones  most  interested.  Ought  we  not  to  consult  their  tastes, 
which  mean  their  capacities?  Their  verdict  is  always  for 
Stickney. 

When  it  is  a  question  of  obstacles,  wings  are  sometimes 
worth  more  than  feet.  Stickney's  Readers  are  inspiring,  and 
lift  the  children  over  difficulties. 

Best  in  idea  and  plan ;  best  in  matter  and  make ;  best  in, 
interest  and  results. 

They  have  found  favor  with  our  teachers  and  pupils  from  the  first.  To 
me  the  books  seem  to  be  just  what  the  gifted  author  intended  them  to  be, 
as  natural  and  beautiful  as  childhood  itself.  They  deserve  the  greatest 
success.  —  A.  R.  Sabin,  Assistant  Supt.,  Chicago,  III. 


GINN  &  COMPANY,  Publishers, 

BOSTON,  NEW  YORK,  AND  CHICAGO. 


RETURN        MARIAN  KOSHLAND  BIOSCIENCE  AND 
TO  ->  NATURAL  RESOURCE  LIBRARY 

2101  Valley  Life  Sciences  Bldg.   642-2531 


ONE 


LOAN 


ALL  BOOKS  MAY  BE  RECALLED  AFTER  7  DAYS. 
DUE  AS  STAMPED  BELOW. 


DUE 


FORM  NO.  DD  8 
24M    4-00 


UNIVERSITY  OF  CALIFORNIA,  BERKELEY 
Berkeley,  California  94720-6500 


YB   17449 


j 

A 


QA  10 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


